Decompositional Extraction and Retrieval of Conceptual Knowledge
An ability to extract hidden and implicit knowledge, their integration into a knowledge base, and then retrieval of required knowledge items are important features of knowledge processing for many modern knowledge-based systems. However, the complexity of these tasks depends on the size of knowledge...
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pp_isofts_kiev_ua-article-5162023-06-25T05:13:02Z Decompositional Extraction and Retrieval of Conceptual Knowledge Декомпозиційне видобування та пошук концептуальних знань Terletskyi, D.O. Yershov, S.V. internal semantic dependencies; decomposition consistency; decomposition of classes; knowledge extraction; knowledge retrieval UDC: 004.82:004.832.2 внутрішні семантичні залежності; консистентність декомпозиці; декомпозиція класів; видобування знань; пошук знань УДК 004.82:004.832.2 An ability to extract hidden and implicit knowledge, their integration into a knowledge base, and then retrieval of required knowledge items are important features of knowledge processing for many modern knowledge-based systems. However, the complexity of these tasks depends on the size of knowledge sources, which were used for extraction, the size of a knowledge base, which is used for the integration of extracted knowledge, as well as the size of a search space, which is used for the retrieval of required knowledge items. Therefore, in this paper, we analyzed the internal semantic dependencies of homogeneous classes of objects and how they affect the decomposition of such classes. Since all subclasses of a homogeneous class of objects form a complete lattice, we applied the methods of formal concept analysis for the knowledge extraction and retrieval within the corresponding concept lattice. We found that such an approach does not consider internal semantic dependencies within a homogeneous class of objects, consequently, it can cause inference and retrieval of formal concepts, which are semantically inconsistent within a modeled domain. We adapted the algorithm for the decomposition of homogeneous classes of objects, within such knowledge representation model as object-oriented dynamic networks, to perform dynamic knowledge extraction and retrieval, adding additional filtration parameters. As the result, the algorithm extracts knowledge via constructing only semantically consistent subclasses of homogeneous classes of objects and then filters them according to the attribute and dependency queries, retrieving knowledge. In addition, we introduced the decomposition consistency coefficient, which allows estimation of how much the algorithm can reduce the search space for knowledge extraction and improves the performance. To demonstrate some possible application scenarios for the improved algorithm, we provided an appropriate example of knowledge extraction and retrieval via decomposition of a particular homogeneous class of objects.Prombles in programming 2022; 3-4: 139-153 Можливості видобувати приховані та неявні знання, інтегрувати їх у базу знань, а потім здійснювати пошук необхідних еле- ментів знань є важливими особливостями обробки знань для багатьох сучасних систем на основі знань. Однак складність цих задач залежить від розміру джерел знань, які використовувалися для видобування, обсягу бази знань, яка використо- вується для інтеграції видобутих знань, а також розміру простору пошуку, який використовується для пошуку необхідних елементів знань. Тому у даній статті ми проаналізували внутрішні семантичні залежності однорідних класів об’єктів і те, як вони впливають на декомпозицію таких класів. Оскільки всі підкласи однорідного класу об’єктів утворюють повну ре- шітку, ми застосували методи аналізу формальних концептів для вилучення та пошуку знань у відповідній концептуальній ґратці. Ми виявили, що такий підхід не враховує внутрішні семантичні залежності в однорідному класі об’єктів, а отже, це може спричинити виведення і пошук формальних понять, які є семантично некоректними у межах галузі знань, що моделюється. Ми адаптували алгоритм декомпозиції однорідних класів об’єктів для такої моделі представлення знань, як об’єктно-орієнтовані динамічні мережі, додавши додаткові параметри фільтрації для динамічного видобування та пошуку знань. У результаті алгоритм видобуває знання шляхом побудови лише семантично коректних підкласів однорідних класів об’єктів, а потім фільтрує їх відповідно до запитів щодо атрибутів та залежностей, виконуючи пошук знань. Крім того, ми ввели коефіцієнт узгодженості декомпозиції, який дозволяє оцінити, наскільки алгоритм може зменшити простір пошуку для видобування знань і покращити продуктивність. Для демонстрації деяких можливих сценаріїв застосування вдоскона- леного алгоритму ми навели відповідний приклад видобування та пошуку знань за допомогою декомпозиції конкретного однорідного класу об’єктів.Prombles in programming 2022; 3-4: 139-153 Інститут програмних систем НАН України 2023-01-23 Article Article application/pdf https://pp.isofts.kiev.ua/index.php/ojs1/article/view/516 10.15407/pp2022.03-04.139 PROBLEMS IN PROGRAMMING; No 3-4 (2022); 139-153 ПРОБЛЕМЫ ПРОГРАММИРОВАНИЯ; No 3-4 (2022); 139-153 ПРОБЛЕМИ ПРОГРАМУВАННЯ; No 3-4 (2022); 139-153 1727-4907 10.15407/pp2022.03-04 uk https://pp.isofts.kiev.ua/index.php/ojs1/article/view/516/569 Copyright (c) 2023 PROBLEMS IN PROGRAMMING |
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internal semantic dependencies decomposition consistency decomposition of classes knowledge extraction knowledge retrieval UDC: 004.82:004.832.2 |
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internal semantic dependencies decomposition consistency decomposition of classes knowledge extraction knowledge retrieval UDC: 004.82:004.832.2 Terletskyi, D.O. Yershov, S.V. Decompositional Extraction and Retrieval of Conceptual Knowledge |
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internal semantic dependencies decomposition consistency decomposition of classes knowledge extraction knowledge retrieval UDC: 004.82:004.832.2 внутрішні семантичні залежності консистентність декомпозиці декомпозиція класів видобування знань пошук знань УДК 004.82:004.832.2 |
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Terletskyi, D.O. Yershov, S.V. |
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Decompositional Extraction and Retrieval of Conceptual Knowledge |
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Decompositional Extraction and Retrieval of Conceptual Knowledge |
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Decompositional Extraction and Retrieval of Conceptual Knowledge |
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Decompositional Extraction and Retrieval of Conceptual Knowledge |
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Decompositional Extraction and Retrieval of Conceptual Knowledge |
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decompositional extraction and retrieval of conceptual knowledge |
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Декомпозиційне видобування та пошук концептуальних знань |
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An ability to extract hidden and implicit knowledge, their integration into a knowledge base, and then retrieval of required knowledge items are important features of knowledge processing for many modern knowledge-based systems. However, the complexity of these tasks depends on the size of knowledge sources, which were used for extraction, the size of a knowledge base, which is used for the integration of extracted knowledge, as well as the size of a search space, which is used for the retrieval of required knowledge items. Therefore, in this paper, we analyzed the internal semantic dependencies of homogeneous classes of objects and how they affect the decomposition of such classes. Since all subclasses of a homogeneous class of objects form a complete lattice, we applied the methods of formal concept analysis for the knowledge extraction and retrieval within the corresponding concept lattice. We found that such an approach does not consider internal semantic dependencies within a homogeneous class of objects, consequently, it can cause inference and retrieval of formal concepts, which are semantically inconsistent within a modeled domain. We adapted the algorithm for the decomposition of homogeneous classes of objects, within such knowledge representation model as object-oriented dynamic networks, to perform dynamic knowledge extraction and retrieval, adding additional filtration parameters. As the result, the algorithm extracts knowledge via constructing only semantically consistent subclasses of homogeneous classes of objects and then filters them according to the attribute and dependency queries, retrieving knowledge. In addition, we introduced the decomposition consistency coefficient, which allows estimation of how much the algorithm can reduce the search space for knowledge extraction and improves the performance. To demonstrate some possible application scenarios for the improved algorithm, we provided an appropriate example of knowledge extraction and retrieval via decomposition of a particular homogeneous class of objects.Prombles in programming 2022; 3-4: 139-153 |
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Інститут програмних систем НАН України |
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2023 |
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139
Моделі і засоби систем баз даних та знань
UDC: 004.82:004.832.2 https://doi.org/10.15407/pp2022.03-04.139
DECOMPOSITIONAL EXTRACTION
AND RETRIEVAL
OF CONCEPTUAL KNOWLEDGE
Dmytro Terletskyi, Sergiy Yershov
An ability to extract hidden and implicit knowledge, their integration into a knowledge base, and then retrieval of required
knowledge items are important features of knowledge processing for many modern knowledge-based systems. However, the
complexity of these tasks depends on the size of knowledge sources, which were used for extraction, the size of a knowledge
base, which is used for the integration of extracted knowledge, as well as the size of a search space, which is used for the
retrieval of required knowledge items. Therefore, in this paper, we analyzed the internal semantic dependencies of homogeneous
classes of objects and how they affect the decomposition of such classes. Since all subclasses of a homogeneous class of objects
form a complete lattice, we applied the methods of formal concept analysis for the knowledge extraction and retrieval within
the corresponding concept lattice. We found that such an approach does not consider internal semantic dependencies within a
homogeneous class of objects, consequently, it can cause inference and retrieval of formal concepts, which are semantically
inconsistent within a modeled domain. We adapted the algorithm for the decomposition of homogeneous classes of objects,
within such knowledge representation model as object-oriented dynamic networks, to perform dynamic knowledge extraction
and retrieval, adding additional filtration parameters. As the result, the algorithm extracts knowledge via constructing only
semantically consistent subclasses of homogeneous classes of objects and then filters them according to the attribute and
dependency queries, retrieving knowledge. In addition, we introduced the decomposition consistency coefficient, which allows
estimation of how much the algorithm can reduce the search space for knowledge extraction and improves the performance. To
demonstrate some possible application scenarios for the improved algorithm, we provided an appropriate example of knowledge
extraction and retrieval via decomposition of a particular homogeneous class of objects.
Keywords: internal semantic dependencies, decomposition consistency, decomposition of classes, knowledge extraction, knowledge
retrieval.
Можливості видобувати приховані та неявні знання, інтегрувати їх у базу знань, а потім здійснювати пошук необхідних еле-
ментів знань є важливими особливостями обробки знань для багатьох сучасних систем на основі знань. Однак складність
цих задач залежить від розміру джерел знань, які використовувалися для видобування, обсягу бази знань, яка використо-
вується для інтеграції видобутих знань, а також розміру простору пошуку, який використовується для пошуку необхідних
елементів знань. Тому у даній статті ми проаналізували внутрішні семантичні залежності однорідних класів об’єктів і те,
як вони впливають на декомпозицію таких класів. Оскільки всі підкласи однорідного класу об’єктів утворюють повну ре-
шітку, ми застосували методи аналізу формальних концептів для вилучення та пошуку знань у відповідній концептуальній
ґратці. Ми виявили, що такий підхід не враховує внутрішні семантичні залежності в однорідному класі об’єктів, а отже,
це може спричинити виведення і пошук формальних понять, які є семантично некоректними у межах галузі знань, що
моделюється. Ми адаптували алгоритм декомпозиції однорідних класів об’єктів для такої моделі представлення знань, як
об’єктно-орієнтовані динамічні мережі, додавши додаткові параметри фільтрації для динамічного видобування та пошуку
знань. У результаті алгоритм видобуває знання шляхом побудови лише семантично коректних підкласів однорідних класів
об’єктів, а потім фільтрує їх відповідно до запитів щодо атрибутів та залежностей, виконуючи пошук знань. Крім того, ми
ввели коефіцієнт узгодженості декомпозиції, який дозволяє оцінити, наскільки алгоритм може зменшити простір пошуку
для видобування знань і покращити продуктивність. Для демонстрації деяких можливих сценаріїв застосування вдоскона-
леного алгоритму ми навели відповідний приклад видобування та пошуку знань за допомогою декомпозиції конкретного
однорідного класу об’єктів.
Ключові слова: внутрішні семантичні залежності, консистентність декомпозиції, декомпозиція класів, видобування знань,
пошук знань.
Introduction
The extraction and retrieval of knowledge are important features of many modern knowledge-based systems.
Such systems are capable to extract new knowledge by analyzing relevant knowledge sources, integrating it with
previously obtained knowledge, and allowing users to search for necessary knowledge items in the knowledge base.
Depending on the chosen knowledge representation model, the extraction of new implicit and hidden knowledge can
be implemented in different ways. For object-oriented knowledge representation models, knowledge extraction can be
performed via universal exploiters of classes, such as union, intersection, difference, and decomposition, which allow
the construction of new classes of objects based on the existed ones.
In this paper, we study the decomposition of homogeneous classes of objects, within such knowledge
representation model as object-oriented dynamic networks, to demonstrate that the algorithm for decomposition of
classes can be used as a tool for knowledge extraction and retrieval. For this purpose, we improved the algorithm
for decomposition of homogeneous classes of objects, which was proposed in [28], by adding more additional
parameters, that allow adaptation of the algorithm, developed for knowledge extraction, to dynamic knowledge
retrieval. We also discovered that classical methods of formal concept analysis do not cover internal semantic
dependencies among properties and methods of homogeneous classes of objects. In addition, we show how the
improved algorithm can reduce the search space during the retrieval of implicit or hidden knowledge, which cannot
be obtained using standard methods of formal concept analysis.
© Д. Терлецький, С. Єршов, 2022
ISSN 1727-4907. Проблеми програмування. 2022. № 3-4. Спеціальний випуск
140
Моделі і засоби систем баз даних та знань
Formal Concepts Analysis
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal
concept analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices.
It provides tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of
two isomorphic complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it
allows inference and retrieval of new concepts within the corresponding formal context. Let us consider the main
concept of the formal concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple (G, M, I), where G is a set of objects of the context, while M is a set
of its attributes, and I is a relation between G and M, which express that an object
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
has an attribute
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
.
Using this definition, any formal context can be represented by a corresponding cross table, where columns
mean attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and
a set of objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
is a set
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
, i.e. all attributes from the set
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
are common for all objects from the set A.
Definition 3. A set of objects with the common attributes
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
is a set
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
, i.e.
all objects from the set
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
have all attributes from the set B.
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context (G, M, I) is a pair (A, B), where
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
is an extent of the
formal concept, while
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
is an its intent, and where
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
.
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to
combination of two ways of set definition, described in [22]. In the first case, a set can be defined by particular
elements (tabular form), while in the second one, it can be determined using the attributes, which must have all
elements of the set (set builder from). In addition, according to [16], the notion of a formal concept is also similar
to a combination of two theoretical forms of class consideration – an extensional and an intensional. From the first
perspective, a class can be defined by the list of its objects, while from the second one it can be defined by the set
of attributes. The definition of the formal concept proposed in [20] combines these two perspectives into a single
notion and provides an opportunity simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
is a set
Моделі і засоби систем баз даних і знань
[Введите текст]
Among the variety of formal systems for the analysis and processing of conceptual knowledge, formal concept
analysis is one of the most developed frameworks, which is based on the mathematical theory of lattices. It provides
tools for the construction, analysis, and processing of conceptual hierarchies, represented in terms of two isomorphic
complete lattices of objects and attributes. Since lattices consist of chains, which are posets, it allows inference and
retrieval of new concepts within the corresponding formal context. Let us consider the main concept of the formal
concept analysis described in [20, 21]. The first step is the definition of the formal context.
Definition 1. A formal context is a tuple ( , , )G M I , where G is a set of objects of the context, while M is a set of its
attributes, and I is a relation between G and M , which express that an object g G has an attribute m M , i.e.
( , )g m I or gIm .
Using this definition, any formal context can be represented by a corresponding cross table, where columns mean
attributes, while rows mean objects. It allows considering a set of common attributes for a set of objects, and a set of
objects that have attributes from a set of common attributes.
Definition 2. A set of common attributes for selected set of objects A G is a set |A m M gIm g A = , i.e. all
attributes from the set A are common for all objects from the set A .
Definition 3. A set of objects with the common attributes B M is a set |B g G gIm m B = , i.e. all objects
from the set B have all attributes from the set B .
Using these notions, we can define a formal concept based on a particular formal context.
Definition 4. A formal concept of the formal context ( , , )G M I is a pair ( , )A B , where A G is an extent of the
formal concept, while B M is an its intent, and where A B = , B A = .
Such definition of the formal concepts, i.e. using the notions of an extent and an intent, is similar to combination of two
ways of set definition, described in [22]. In the first case, a set can be defined by particular elements (tabular form),
while in the second one, it can be determined using the attributes, which must have all elements of the set (set builder
from). In addition, according to [16], the notion of a formal concept is also similar to a combination of two theoretical
forms of class consideration – an extensional and an intensional. From the first perspective, a class can be defined by
the list of its objects, while from the second one it can be defined by the set of attributes. The definition of the formal
concept proposed in [20] combines these two perspectives into a single notion and provides an opportunity
simultaneously to consider a particular formal concept using both of them.
Since a formal context can define a certain number of formal concepts, there is a sense to define a set of all
formal concepts.
Definition 5. A set of all formal concepts of the formal context ( , , )G M I is a set ( , , )PS G M I .
Formal concept analysis has different applications within an area of knowledge processing. According to [31],
conceptual knowledge retrieval is one of the main categories among the variety of methods of formal knowledge
processing. On another side, these methods allow the implementation of corresponding functionality within knowledge-
based systems developed based on formal concept analysis. In general, the knowledge retrieval task can be simply
described as querying a knowledge base to find the required knowledge items. According to [7, 9-11, 15, 17, 18, 31],
the formal concept analysis allows defining a formal context, where the intent of the context is defined by pieces of
knowledge, for example, keywords or part of sentences, while the extent is defined by the list of documents, that
contain or do not contain such knowledge items. The corresponding concept lattice, constructed based on the formal
context, describes the search space, consequently, the retrieval process can be interpreted as the matching of the search
query with the formal concepts, which are represented by lattice nodes, using different search strategies based on the
relations of generalization and specialization defined between formal concepts. The performance of the retrieval process
depends on the size of the search space and the corresponding search strategy. Therefore, as was noted in [32], one of
the main goals for many retrieval algorithms is to reduce the search space as much as possible. Another issue related to
query matching is the correspondence level of each formal concept to the query, as it can be rather partial than
complete. Thus, in the many search strategies queries are described in a form of inclusion conditions, which allow the
handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to studying
dependencies within the procedural program constructions. One of such was proposed in [30], the main idea of which is
to construct the concept lattice of decomposition slices of the program. It provides an opportunity to analyze groups of
the ordered program statements, called decomposition slices, related to a particular context, for example to a variable. In
other words, each particular variable, which is a part of the program, depends on the corresponding ordered sequence of
operators, which somehow use or change its state. However, the proposed approach was designed for procedural
.
Formal concept analysis has different applications within an area of knowledge processing. According
to [31], conceptual knowledge retrieval is one of the main categories among the variety of methods of formal
knowledge processing. On another side, these methods allow the implementation of corresponding functionality
within knowledge-based systems developed based on formal concept analysis. In general, the knowledge retrieval
task can be simply described as querying a knowledge base to find the required knowledge items. According to [7,
9-11, 15, 17, 18, 31], the formal concept analysis allows defining a formal context, where the intent of the context is
defined by pieces of knowledge, for example, keywords or part of sentences, while the extent is defined by the list
of documents, that contain or do not contain such knowledge items. The corresponding concept lattice, constructed
based on the formal context, describes the search space, consequently, the retrieval process can be interpreted as
the matching of the search query with the formal concepts, which are represented by lattice nodes, using different
search strategies based on the relations of generalization and specialization defined between formal concepts. The
performance of the retrieval process depends on the size of the search space and the corresponding search strategy.
Therefore, as was noted in [32], one of the main goals for many retrieval algorithms is to reduce the search space as
much as possible. Another issue related to query matching is the correspondence level of each formal concept to the
query, as it can be rather partial than complete. Thus, in the many search strategies queries are described in a form
of inclusion conditions, which allow the handling of partial query matching within the concept lattice.
Morphology of Classes
Nowadays, there are a few approaches, which propose the application of the formal concept analysis to
studying dependencies within the procedural program constructions. One of such was proposed in [30], the main
idea of which is to construct the concept lattice of decomposition slices of the program. It provides an opportunity
to analyze groups of the ordered program statements, called decomposition slices, related to a particular context,
for example to a variable. In other words, each particular variable, which is a part of the program, depends on
the corresponding ordered sequence of operators, which somehow use or change its state. However, the proposed
approach was designed for procedural programs, but not for object-oriented ones. Consequently, it is more suitable
for the structure analysis of procedural knowledge than for the analysis of declarative knowledge represented in
terms of classes. A similar approach, but for the analysis of class methods cohesion, was proposed in [29]. The
main idea is to consider dependencies between different program statements within a particular method of a class
and define the corresponding formal context using them, and then construct the concept lattice called a cohesion
lattice. However, the approach does not pay the attention to the external dependencies of class attributes used
in the method with other properties and methods of the class, which are important for the decomposition of
homogeneous classes of objects.
141
Моделі і засоби систем баз даних та знань
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using
a set of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize
its structure. The embedded call graph provides additional information about the interaction between methods of the
class, which is absent in the corresponding concept lattice. However, such an approach covers only dependencies
between methods of the class and does not pay the attention to other kinds of dependencies, for example, between
properties, properties and methods of the class. Another application of the class formal context was proposed in [23],
which was used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion
lattice. They capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the
class structure more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures
of dependencies among class members, than qualitative. Many approaches to class cohesion measurements pay the
attention to dependencies’ existence but not to their semantics and consistency within a modeled domain, which is
crucial for the decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is
the detection of functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional
dependency is defined as the implication over the relation pairs, determined on the set of attributes, which
are mapped into columns of particular tables. The main idea of functional dependencies is to conclude that if
two particular tuples of attributes in the relation contain a certain attribute X, which is called an antecedent,
then they also contain another attribute Y, which is called a consequent. Such facts can be considered new
knowledge, which is hidden or implicit. In addition, there is a generalized form of functional dependency
called a similarity dependency [4-6, 8], the main idea of which is the satisfaction of functional dependency
for any two tuples in the relation. However, such kinds of dependencies do not cover the internal semantic
connection within classes and objects because they consider only the availability of a particular attribute for
an object, rather how the different attributes of the object are related to each other, or more precisely, how
they depend on each other. They do not consider how the presence or absence of one particular attribute for an
object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
depends on another attribute
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
, whenever the presence of
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
is
not significant without the presence of
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
, where
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
and
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
, where
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
is a
collection of properties which define the structure of the class T, while
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
is a collection of its methods,
that define its behavior.
Definition 7. A homogeneous class of objects Ti is a subclass of homogeneous class of objects T, i.e.
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
, if
and only if
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
where
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
are specifications and signatures
of the class
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
and T, respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to
understand how the properties and methods can depend on each other, creating internal semantic dependencies. For
this purpose, let us define a homogeneous class of objects Pt, which describes a concept of a point on a plane, and has
the following structure:
Моделі і засоби систем баз даних і знань
programs, but not for object-oriented ones. Consequently, it is more suitable for the structure analysis of procedural
knowledge than for the analysis of declarative knowledge represented in terms of classes. A similar approach, but for
the analysis of class methods cohesion, was proposed in [29]. The main idea is to consider dependencies between
different program statements within a particular method of a class and define the corresponding formal context using
them, and then construct the concept lattice called a cohesion lattice. However, the approach does not pay the attention
to the external dependencies of class attributes used in the method with other properties and methods of the class, which
are important for the decomposition of homogeneous classes of objects.
Usually, a formal context is defined using a set of attributes and a set of objects, where attributes have
corresponding values encapsulated in a particular object. However, a formal context can also be determined using a set
of classes and a set of attributes. This idea was used in [13, 14] to analyze the structure of classes in the Java
programming language, in particular, to consider the interrelation between methods call of a class and then optimize its
structure. The embedded call graph provides additional information about the interaction between methods of the class,
which is absent in the corresponding concept lattice. However, such an approach covers only dependencies between
methods of the class and does not pay the attention to other kinds of dependencies, for example, between properties,
properties and methods of the class. Another application of the class formal context was proposed in [23], which was
used to analyze the class cohesion via the construction of the corresponding concept lattice called cohesion lattice. They
capture the cohesiveness of a class and its members, which provides an opportunity to reorganize the class structure
more efficiently, increasing cohesion. However, cohesion metrics are rather quantitative measures of dependencies
among class members, than qualitative. Many approaches to class cohesion measurements pay the attention to
dependencies’ existence but not to their semantics and consistency within a modeled domain, which is crucial for the
decomposition of homogeneous classes.
One of the known approaches, that considers dependencies between the attributes of an object, is the detection of
functional dependencies in relational databases. As it was noted in [4-6, 8, 32], a functional dependency is defined as
the implication over the relation pairs, determined on the set of attributes, which are mapped into columns of particular
tables. The main idea of functional dependencies is to conclude that if two particular tuples of attributes in the relation
contain a certain attribute X , which is called an antecedent, then they also contain another attribute Y , which is called
a consequent. Such facts can be considered new knowledge, which is hidden or implicit. In addition, there is a
generalized form of functional dependency called a similarity dependency [4-6, 8], the main idea of which is the
satisfaction of functional dependency for any two tuples in the relation. However, such kinds of dependencies do not
cover the internal semantic connection within classes and objects because they consider only the availability of a
particular attribute for an object, rather how the different attributes of the object are related to each other, or more
precisely, how they depend on each other. They do not consider how the presence or absence of one particular attribute
for an object or a class affects their semantic consistency.
An alternative approach to the analysis of dependencies between attributes of objects was proposed in [25],
according to which an attribute 2m depends on another attribute 1m , i.e. 1 2m m , whenever the presence of 2m is not
significant without the presence of 1m , where 1m and 2m also may be atomic, as well as conjunctive or disjunctive
attributes. However, such an explanation of the dependency between attributes is quite fuzzy because it is unclear how
to verify that presence of one attribute is not significant without the presence of another one, as well as what the term
significance should be meant here.
To consider what internal semantic dependencies of a class are, their kinds, and how they affect the
decomposition of the homogeneous classes of objects, let us consider the definitions of a homogeneous class of objects
and its subclass within such knowledge representation model as object-oriented dynamic networks (OODNs), which
was proposed in [26, 27].
Definition 6. The homogeneous class of objects T is a tuple ( ) ( )( ),T P T F T= , where ( ) ( ) ( )( )1 ,..., nP T p T p T= is a
collection of properties which define the structure of the class T , while ( ) ( ) ( )( )1 ,..., mF T f T f T= is a collection of its
methods, that define its behavior.
Definition 7. A homogeneous class of objects iT is a subclass of homogeneous class of objects T , i.e. iT T , if and
only if ( ) ( )iP T P T and ( ) ( )iF T F T , where ( )iP T , ( )P T and ( )iF T , ( )F T are specifications and signatures
of the class iT and T , respectively.
Let us consider an example of a homogeneous class of objects and analyze its specification and signature to understand
how the properties and methods can depend on each other, creating internal semantic dependencies. For this purpose, let
us define a homogeneous class of objects Pt , which describes a concept of a point on a plane, and has the following
structure:
( )( )( ( )( ) ( ) ( )
( )( ) ( )( ))
1 2 1 2
3 4
, , , , , , _ , , _ , ,
_ , , , _ , , ,
x yPt p x v p y v f get x pt f get y pt
f set x pt x f set y pt y
= = = =
= =
where
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
and
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
are quantitative properties, which mean coordinates
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
of a point
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
and
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
are
methods, which return x and y coordinates of a point pt, respectively;
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
and
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
are methods, which provide an
opportunity to set a value of x and y coordinates of a point pt, respectively.
Now let us define another homogeneous class of objects Tr, which describes a concept of a triangle on a plane,
and has the following structure:
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
where
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
are quantitative properties, which mean vertices of a triangle, defined as objects of
the class
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
142
Моделі і засоби систем баз даних та знань
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
where
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
are defined as follows:
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a method, which returns the coordinates
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a method, that set coordinates
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
for a vertex n of a
triangle tr and is defined as follows
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a method, which
returns a distance between
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
and
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
of a triangle tr, and defined as follows:
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a method, which returns the perimeter of a triangle tr and is defined as follows
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
,
where s1, s2, and s3 are defined in the same way as in for the
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
.
Let us consider the class Tr as a collection of properties and methods, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
.
We denote an i-th property of the class
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
for a more compact
representation of all statements noted below.
As it was noted in [32], for the class Tr we can construct
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
subclasses, which create the power
set lattice
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
, which is a complete lattice and where
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a set of all possible unique subsets
of the set
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
. From the decomposition perspective, we need to consider only
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
subclass, which are
nonempty ones and create the join-semilattice
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
, which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous
class of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses
of the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context W1. We use the symbol plus + to specify the pair of the relation
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
. An object g is defined as follows
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a j-th subclass of the class Tr, which has a cardinality of
i and
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M .
is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M, which have
a cardinality i; and where an attribute (property or method) m is defined as
Моделі і засоби систем баз даних і знань
[Введите текст]
where 1.Pt p and 2.Pt p are quantitative properties, which mean coordinates ( ),x y of a point pt ; 1.Pt f and 2.Pt f are
methods, which return x and y coordinates of a point pt , respectively; 3.Pt f and 4.Pt f are methods, which provide
an opportunity to set a value of x and y coordinates of a point pt , respectively.
Now let us define another homogeneous class of objects Tr , which describes a concept of a triangle on a plane,
and has the following structure:
( )( ) ( )( ) ( )( ) ( ) ( )(
( )( ) ( ) ( ) ( )( )
( ) ( )( )
1 1 1 2 2 2 3 3 3 4 4
1 2
3 4
, , , , , , , , , _ _ , 0,1 ,
_ , , , , _ , , , , , , ,
_ _ , , , , , , _
x y
a b
Tr p vertex v Pt p vertex v Pt p vertex v Pt p is a triangle vf tr v
f get vertex tr n Pt f set vertex tr n a a
f get side length tr vertex Pt vertex Pt f get perimeter tr
+ +
+
= = = =
= =
= = ( )), ,R+
where 1.Tr p , 2.Tr p , and 3.Tr p are quantitative properties, which mean vertices of a triangle, defined as objects of the
class Pt ; 4.Tr p is a qualitative property, which means satisfiability of the triangle inequality and is defined by the
following verification function:
( ) ( ) ( ) ( ) ( )( )4 1 2 3 4 1 2 3 1 3 2 2 3 1: . , . , . 0,1 , ,vf tr tr vertex tr vertex tr vertex vf s s s s s s s s s→ = + + +
where 1s , 2s , and 3s are defined as follows:
( ) ( )( )
( ) ( )( )
( ) ( )( )
1
2
3
_ _ . _ 1 , . _ 2 ,
_ _ . _ 1 , . _ 3 ,
_ _ . _ 2 , . _ 3 ,
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
s get side length tr get vertex tr get vertex
=
=
=
1.Tr f is a method, which returns the coordinates ( ),x y for a vertex n of a triangle tr in a form of objects of the class
Pt and is defined as follows ( ) ( )1 , . ;nf tr n tr vertex= 2.Tr f is a method, that set coordinates ( ),x ya a for a vertex n of
a triangle tr and is defined as follows ( ) ( ) ( )( )2 , , , . . _ , . . _ ;x y n x n yf tr n a a tr vertex set x a tr vertex set y a= 3.Tr f is a
method, which returns a distance between avertex and bvertex of a triangle tr , and defined as follows:
( ) ( ) ( )2 2
3 , , . _ () . _ () . _ () . _ () ;a b a b a bf tr vertex vertex vertex get x vertex get x vertex get y vertex get y= − + −
4.Tr f is a method, which returns the perimeter of a triangle tr and is defined as follows ( )4 1 2 3f tr s s s= + + , where 1s ,
2s , and 3s are defined in the same way as in for the ( )4vf tr .
Let us consider the class Tr as a collection of properties and methods, i.e.
( ) ( ) 1 2 3 4 1 2 3 4. , . , . , . , . , . , . , .Tr P Tr F Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f Tr f= = .
We denote an i -th property of the class Tr by . iTr p , and a j -th method – by . jTr f for a more compact representation
of all statements noted below.
As it was noted in [32], for the class Tr we can construct 82 2 256n = = subclasses, which create the power set
lattice ( )( ), , ,L PS Tr= , which is a complete lattice and where ( )PS Tr is a set of all possible unique subsets of
the set ( ) ( )P Tr F Tr . From the decomposition perspective, we need to consider only 2 1n − subclass, which are
nonempty ones and create the join-semilattice ( ) ( )\ , ,JSL PS Tr= , which describes the search space for the
knowledge retrieval. This semilattice is not a concept lattice, however as it was demonstrated in [28], it can crucially
reduce the space search for the solving decomposition constraint satisfaction problem.
To compare the opportunities provided by the join-semilattice of nonempty subclasses of the homogeneous class
of objects Tr and the concept lattice of all its subclasses, let us define the formal context of all possible subclasses of
the class Tr and then construct the corresponding concept lattice. For this purpose, let us consider the formal context
( ) ( ) ( )( )1 , , :W G PS Tr M P Tr F Tr I G M= = = and define the corresponding cross table for it. We do not provide
a full cross table here because of its size, however, Table 1 illustrates the basic intuition for the definition of formal
context 1W . We use the symbol plus + to specify the pair of the relation ( ),g m I . An object g is defined as follows
( )i
jg SC Tr Tr , 0,| |i M= , | |1, i
Mj C= , where ( )i
jSC Tr is a j -th subclass of the class Tr , which has a cardinality
of i and | |
i
TrC is a binomial coefficient, which is equal to a number of all possible unique subsets of the set M , which
have a cardinality i ; and where an attribute (property or method) m is defined as m M . .
Now, let us construct the concept lattice for the formal context W1 using the Table 1. Analyzing the
results, depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all
formally possible subclasses of the class Tr and all possible formal concepts of the context W1. Considering the
constructed concept lattice, we can ask a question about the semantic consistency of all constructed subclasses of
the class Tr, as it was done in [28]. To clarify the problem and then answer this question, let us consider in more
detail the internal structure of the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr.
W
Properties and methods
Tr.p1 Tr.p2 Tr.p3 Tr.p4 Tr.f1 Tr.f2 Tr.f3 Tr.f4
Su
bc
la
ss
es
SC1
0 (Tr)
SC1
1 (Tr) +
SC2
1 (Tr) +
... ...
SC1
8 (Tr) + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties
Моделі і засоби систем баз даних і знань
Now, let us construct the concept lattice for the formal context 1W using the Table 1. Analyzing the results,
depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all formally possible
subclasses of the class Tr and all possible formal concepts of the context 1W . Considering the constructed concept
lattice, we can ask a question about the semantic consistency of all constructed subclasses of the class Tr , as it was
done in [28]. To clarify the problem and then answer this question, let us consider in more detail the internal structure of
the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr .
W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
Su
bc
la
ss
es
( )0
1SC Tr
( )1
1SC Tr +
( )1
2SC Tr +
... ...
( )8
1SC Tr + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties ( ) 1 2 3 4. , . , . , .P Tr Tr p Tr p Tr p Tr p= , and a collection of methods ( ) 1 2 3 4. , . , . , .F Tr Tr f Tr f Tr f Tr f= called
a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr , we can
find some internal dependencies among them. It is a common practice for many object-oriented programming languages
as well as knowledge representation models to define some properties and methods of a class using for this purpose
other properties and (or) methods of the class. Such practice allows us to avoid code duplication and provides instead of
it code reusability. However, it creates internal dependencies, which help to describe the modeled instance more
precisely to the corresponding entity from a particular domain. In addition, such dependencies are important for the
decomposition of classes because they define appropriate constraints for the properties and methods of a class.
Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only some of them do
not conflict with constraints imposed by the dependencies, the decomposition of a class as the construction of all its
subclasses is based on such dependencies.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr , to understand
the problem more specifically. One of the semantically consistent subclasses of the class Tr is the subclass
3
16 1 1 2. , . , .SC Tr p Tr f Tr f= , because the property 1.Tr p is defined independently from other properties and methods,
and it is required for the execution of methods 1.Tr f and 2.Tr f , which are determined based on this property. In other
Figure 1. Concept lattice of the formal context 1W .
, and a collection of methods
Моделі і засоби систем баз даних і знань
Now, let us construct the concept lattice for the formal context 1W using the Table 1. Analyzing the results,
depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all formally possible
subclasses of the class Tr and all possible formal concepts of the context 1W . Considering the constructed concept
lattice, we can ask a question about the semantic consistency of all constructed subclasses of the class Tr , as it was
done in [28]. To clarify the problem and then answer this question, let us consider in more detail the internal structure of
the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr .
W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
Su
bc
la
ss
es
( )0
1SC Tr
( )1
1SC Tr +
( )1
2SC Tr +
... ...
( )8
1SC Tr + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties ( ) 1 2 3 4. , . , . , .P Tr Tr p Tr p Tr p Tr p= , and a collection of methods ( ) 1 2 3 4. , . , . , .F Tr Tr f Tr f Tr f Tr f= called
a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr , we can
find some internal dependencies among them. It is a common practice for many object-oriented programming languages
as well as knowledge representation models to define some properties and methods of a class using for this purpose
other properties and (or) methods of the class. Such practice allows us to avoid code duplication and provides instead of
it code reusability. However, it creates internal dependencies, which help to describe the modeled instance more
precisely to the corresponding entity from a particular domain. In addition, such dependencies are important for the
decomposition of classes because they define appropriate constraints for the properties and methods of a class.
Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only some of them do
not conflict with constraints imposed by the dependencies, the decomposition of a class as the construction of all its
subclasses is based on such dependencies.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr , to understand
the problem more specifically. One of the semantically consistent subclasses of the class Tr is the subclass
3
16 1 1 2. , . , .SC Tr p Tr f Tr f= , because the property 1.Tr p is defined independently from other properties and methods,
and it is required for the execution of methods 1.Tr f and 2.Tr f , which are determined based on this property. In other
Figure 1. Concept lattice of the formal context 1W .
called a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr,
we can find some internal dependencies among them. It is a common practice for many object-oriented programming
languages as well as knowledge representation models to define some properties and methods of a class using for
this purpose other properties and (or) methods of the class. Such practice allows us to avoid code duplication and
provides instead of it code reusability. However, it creates internal dependencies, which help to describe the modeled
143
Моделі і засоби систем баз даних та знань
instance more precisely to the corresponding entity from a particular domain. In addition, such dependencies are
important for the decomposition of classes because they define appropriate constraints for the properties and methods
of a class. Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only
some of them do not conflict with constraints imposed by the dependencies, the decomposition of a class as the
construction of all its subclasses is based on such dependencies.
Моделі і засоби систем баз даних і знань
Now, let us construct the concept lattice for the formal context 1W using the Table 1. Analyzing the results,
depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all formally possible
subclasses of the class Tr and all possible formal concepts of the context 1W . Considering the constructed concept
lattice, we can ask a question about the semantic consistency of all constructed subclasses of the class Tr , as it was
done in [28]. To clarify the problem and then answer this question, let us consider in more detail the internal structure of
the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr .
W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
Su
bc
la
ss
es
( )0
1SC Tr
( )1
1SC Tr +
( )1
2SC Tr +
... ...
( )8
1SC Tr + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties ( ) 1 2 3 4. , . , . , .P Tr Tr p Tr p Tr p Tr p= , and a collection of methods ( ) 1 2 3 4. , . , . , .F Tr Tr f Tr f Tr f Tr f= called
a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr , we can
find some internal dependencies among them. It is a common practice for many object-oriented programming languages
as well as knowledge representation models to define some properties and methods of a class using for this purpose
other properties and (or) methods of the class. Such practice allows us to avoid code duplication and provides instead of
it code reusability. However, it creates internal dependencies, which help to describe the modeled instance more
precisely to the corresponding entity from a particular domain. In addition, such dependencies are important for the
decomposition of classes because they define appropriate constraints for the properties and methods of a class.
Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only some of them do
not conflict with constraints imposed by the dependencies, the decomposition of a class as the construction of all its
subclasses is based on such dependencies.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr , to understand
the problem more specifically. One of the semantically consistent subclasses of the class Tr is the subclass
3
16 1 1 2. , . , .SC Tr p Tr f Tr f= , because the property 1.Tr p is defined independently from other properties and methods,
and it is required for the execution of methods 1.Tr f and 2.Tr f , which are determined based on this property. In other
Figure 1. Concept lattice of the formal context 1W . Figure 1. Concept lattice of the formal context W1.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr, to
understand the problem more specifically. One of the semantically consistent subclasses of the class Tr is the
subclass
Моделі і засоби систем баз даних і знань
Now, let us construct the concept lattice for the formal context 1W using the Table 1. Analyzing the results,
depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all formally possible
subclasses of the class Tr and all possible formal concepts of the context 1W . Considering the constructed concept
lattice, we can ask a question about the semantic consistency of all constructed subclasses of the class Tr , as it was
done in [28]. To clarify the problem and then answer this question, let us consider in more detail the internal structure of
the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr .
W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
Su
bc
la
ss
es
( )0
1SC Tr
( )1
1SC Tr +
( )1
2SC Tr +
... ...
( )8
1SC Tr + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties ( ) 1 2 3 4. , . , . , .P Tr Tr p Tr p Tr p Tr p= , and a collection of methods ( ) 1 2 3 4. , . , . , .F Tr Tr f Tr f Tr f Tr f= called
a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr , we can
find some internal dependencies among them. It is a common practice for many object-oriented programming languages
as well as knowledge representation models to define some properties and methods of a class using for this purpose
other properties and (or) methods of the class. Such practice allows us to avoid code duplication and provides instead of
it code reusability. However, it creates internal dependencies, which help to describe the modeled instance more
precisely to the corresponding entity from a particular domain. In addition, such dependencies are important for the
decomposition of classes because they define appropriate constraints for the properties and methods of a class.
Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only some of them do
not conflict with constraints imposed by the dependencies, the decomposition of a class as the construction of all its
subclasses is based on such dependencies.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr , to understand
the problem more specifically. One of the semantically consistent subclasses of the class Tr is the subclass
3
16 1 1 2. , . , .SC Tr p Tr f Tr f= , because the property 1.Tr p is defined independently from other properties and methods,
and it is required for the execution of methods 1.Tr f and 2.Tr f , which are determined based on this property. In other
Figure 1. Concept lattice of the formal context 1W .
, because the property
Моделі і засоби систем баз даних і знань
Now, let us construct the concept lattice for the formal context 1W using the Table 1. Analyzing the results,
depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all formally possible
subclasses of the class Tr and all possible formal concepts of the context 1W . Considering the constructed concept
lattice, we can ask a question about the semantic consistency of all constructed subclasses of the class Tr , as it was
done in [28]. To clarify the problem and then answer this question, let us consider in more detail the internal structure of
the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr .
W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
Su
bc
la
ss
es
( )0
1SC Tr
( )1
1SC Tr +
( )1
2SC Tr +
... ...
( )8
1SC Tr + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties ( ) 1 2 3 4. , . , . , .P Tr Tr p Tr p Tr p Tr p= , and a collection of methods ( ) 1 2 3 4. , . , . , .F Tr Tr f Tr f Tr f Tr f= called
a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr , we can
find some internal dependencies among them. It is a common practice for many object-oriented programming languages
as well as knowledge representation models to define some properties and methods of a class using for this purpose
other properties and (or) methods of the class. Such practice allows us to avoid code duplication and provides instead of
it code reusability. However, it creates internal dependencies, which help to describe the modeled instance more
precisely to the corresponding entity from a particular domain. In addition, such dependencies are important for the
decomposition of classes because they define appropriate constraints for the properties and methods of a class.
Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only some of them do
not conflict with constraints imposed by the dependencies, the decomposition of a class as the construction of all its
subclasses is based on such dependencies.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr , to understand
the problem more specifically. One of the semantically consistent subclasses of the class Tr is the subclass
3
16 1 1 2. , . , .SC Tr p Tr f Tr f= , because the property 1.Tr p is defined independently from other properties and methods,
and it is required for the execution of methods 1.Tr f and 2.Tr f , which are determined based on this property. In other
Figure 1. Concept lattice of the formal context 1W .
is defined independently from other properties
and methods, and it is required for the execution of methods
Моделі і засоби систем баз даних і знань
Now, let us construct the concept lattice for the formal context 1W using the Table 1. Analyzing the results,
depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all formally possible
subclasses of the class Tr and all possible formal concepts of the context 1W . Considering the constructed concept
lattice, we can ask a question about the semantic consistency of all constructed subclasses of the class Tr , as it was
done in [28]. To clarify the problem and then answer this question, let us consider in more detail the internal structure of
the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr .
W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
Su
bc
la
ss
es
( )0
1SC Tr
( )1
1SC Tr +
( )1
2SC Tr +
... ...
( )8
1SC Tr + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties ( ) 1 2 3 4. , . , . , .P Tr Tr p Tr p Tr p Tr p= , and a collection of methods ( ) 1 2 3 4. , . , . , .F Tr Tr f Tr f Tr f Tr f= called
a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr , we can
find some internal dependencies among them. It is a common practice for many object-oriented programming languages
as well as knowledge representation models to define some properties and methods of a class using for this purpose
other properties and (or) methods of the class. Such practice allows us to avoid code duplication and provides instead of
it code reusability. However, it creates internal dependencies, which help to describe the modeled instance more
precisely to the corresponding entity from a particular domain. In addition, such dependencies are important for the
decomposition of classes because they define appropriate constraints for the properties and methods of a class.
Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only some of them do
not conflict with constraints imposed by the dependencies, the decomposition of a class as the construction of all its
subclasses is based on such dependencies.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr , to understand
the problem more specifically. One of the semantically consistent subclasses of the class Tr is the subclass
3
16 1 1 2. , . , .SC Tr p Tr f Tr f= , because the property 1.Tr p is defined independently from other properties and methods,
and it is required for the execution of methods 1.Tr f and 2.Tr f , which are determined based on this property. In other
Figure 1. Concept lattice of the formal context 1W .
and
Моделі і засоби систем баз даних і знань
Now, let us construct the concept lattice for the formal context 1W using the Table 1. Analyzing the results,
depicted in Figure 1, we can see, that the constructed concept lattice has a big size and contains all formally possible
subclasses of the class Tr and all possible formal concepts of the context 1W . Considering the constructed concept
lattice, we can ask a question about the semantic consistency of all constructed subclasses of the class Tr , as it was
done in [28]. To clarify the problem and then answer this question, let us consider in more detail the internal structure of
the class Tr and how it is related to the semantic consistency of its subclasses.
Table 1. Formal context, which defines all possible subclasses of the class Tr .
W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
Su
bc
la
ss
es
( )0
1SC Tr
( )1
1SC Tr +
( )1
2SC Tr +
... ...
( )8
1SC Tr + + + + + + + +
According to the definition of the homogeneous class of objects, the class Tr consists of a collection of
properties ( ) 1 2 3 4. , . , . , .P Tr Tr p Tr p Tr p Tr p= , and a collection of methods ( ) 1 2 3 4. , . , . , .F Tr Tr f Tr f Tr f Tr f= called
a specification and a signature respectively. Analyzing definitions of properties and methods of the class Tr , we can
find some internal dependencies among them. It is a common practice for many object-oriented programming languages
as well as knowledge representation models to define some properties and methods of a class using for this purpose
other properties and (or) methods of the class. Such practice allows us to avoid code duplication and provides instead of
it code reusability. However, it creates internal dependencies, which help to describe the modeled instance more
precisely to the corresponding entity from a particular domain. In addition, such dependencies are important for the
decomposition of classes because they define appropriate constraints for the properties and methods of a class.
Furthermore, since not all formally possible subclasses of a class are semantically consistent, i.e., only some of them do
not conflict with constraints imposed by the dependencies, the decomposition of a class as the construction of all its
subclasses is based on such dependencies.
Let us consider examples of semantically consistent and inconsistent subclasses of the class Tr , to understand
the problem more specifically. One of the semantically consistent subclasses of the class Tr is the subclass
3
16 1 1 2. , . , .SC Tr p Tr f Tr f= , because the property 1.Tr p is defined independently from other properties and methods,
and it is required for the execution of methods 1.Tr f and 2.Tr f , which are determined based on this property. In other
Figure 1. Concept lattice of the formal context 1W .
, which are determined based on this
property. In other words, subclass
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
defines a point on a plane with the ability to get and set its coordinates.
One of the semantically inconsistent subclass of the class Tr is a subclass the
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
because
as in the previous case, the property
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
is defined independently from other properties and methods, and it is
required for the invocation of the method
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
, however, the correct invocation of the method
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
demands one
more property similar to
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
. In other words, subclass
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
defines a point on a plane with the ability to get its
coordinates, but the invocation of the method
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
, which computes the distance between two points on a plane,
will cause an error because the class
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
determines only one point on a plane. Therefore, the subclass
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
is
inconsistent one. Using this fact, we can conclude that the constructed concept lattice of the formal context W1
contains semantically consistent concepts, as well as inconsistent ones. This fact is important for knowledge
retrieval since it is avoiding the consideration of semantically inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well
as structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since
properties and methods of a class can be defined independently of other properties and (or) methods of the class,
as well as using them, they are similar to chemical atoms and molecules. Indeed, independent properties and
methods are similar to atoms, which are the smallest indivisible particles, while dependent ones are similar to
molecules, which are groups of atoms or smaller molecules somehow connected with each other. As the properties
of a class define its structure, while the methods define its behavior, the corresponding atoms and molecules of
the class can be classified as structural and functional ones. Let us consider the definitions of both kinds of atoms
and molecules of a class, as well as some of their examples, using the class Tr.
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
,
where
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
is a property defined without using any other properties and (or) methods of the class T, where P(T)
is its specification.
144
Моделі і засоби систем баз даних та знань
To analyze the specification of the class Tr, we can find, that quantitative properties
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
, and
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
which mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr.
Therefore, these properties define structural atoms
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
of the class Tr, respectively, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
,
where
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
is a method defined without using any other properties and (or) methods of the class T, where F(T)
is its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
is a method defined based on the other methods and (or) properties
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
which form structural and (or) functional atoms, and are parts of smaller molecules of
the class T, where
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
is a specification of the class of objects T, while T(F) is
its signature.
To analyze the structure and behavior of the class Tr, we can observe that methods
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
, which get
and set the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
of the class Tr, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
In addition, method
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
, which computes the length of a particular side of a triangle, uses a corresponding
pair of its vertices. Therefore, it determines a functional molecule
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
of the class Tr, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
words, subclass 3
16SC defines a point on a plane with the ability to get and set its coordinates. One of the semantically
inconsistent subclass of the class Tr is a subclass the 3
17 1 1 3. , . , .SC Tr p Tr f Tr f= because as in the previous case, the
property 1.Tr p is defined independently from other properties and methods, and it is required for the invocation of the
method 1.Tr f , however, the correct invocation of the method 3.Tr f demands one more property similar to 1.Tr p . In
other words, subclass 3
17SC defines a point on a plane with the ability to get its coordinates, but the invocation of the
method 3.Tr f , which computes the distance between two points on a plane, will cause an error because the class 3
17SC
determines only one point on a plane. Therefore, the subclass 3
17SC is inconsistent one. Using this fact, we can conclude
that the constructed concept lattice of the formal context 1W contains semantically consistent concepts, as well as
inconsistent ones. This fact is important for knowledge retrieval since it is avoiding the consideration of semantically
inconsistent subclasses and reduce the search space.
To formalize the internal dependencies of a class, concepts of structural and functional atoms, as well as
structural and functional molecules of the homogeneous class of objects, were introduced in [28]. Since properties and
methods of a class can be defined independently of other properties and (or) methods of the class, as well as using them,
they are similar to chemical atoms and molecules. Indeed, independent properties and methods are similar to atoms,
which are the smallest indivisible particles, while dependent ones are similar to molecules, which are groups of atoms
or smaller molecules somehow connected with each other. As the properties of a class define its structure, while the
methods define its behavior, the corresponding atoms and molecules of the class can be classified as structural and
functional ones. Let us consider the definitions of both kinds of atoms and molecules of a class, as well as some of their
examples, using the class Tr .
Definition 8. Structural atom of a homogeneous class of objects T is a singleton collection ( ) .i iSA T T p= , where
( ). iT p P T is a property defined without using any other properties and (or) methods of the class T , where ( )P T is
its specification.
To analyze the specification of the class Tr , we can find, that quantitative properties 1.Tr p , 2.Tr p , and 3.Tr p , which
mean the vertices of a triangle, are defined without usage of any other property or method of the class Tr . Therefore,
these properties define structural atoms ( )1SA Tr , ( )2SA Tr , and ( )3SA Tr of the class Tr , respectively, i.e.
( ) ( ) ( ) 1 1 2 2 3 3. , . , . .SA Tr Tr p SA Tr Tr p SA Tr Tr p= = =
Definition 9. Functional atom of a homogeneous class of objects T is singleton collection ( ) .i iFA T T f= , where
( ). iT f F T is a method defined without using any other properties and (or) methods of the class T , where ( )F T is
its signature.
The signature of class Tr does not contain any methods defined independently from other properties and methods.
Definition 10. A functional molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jFM T T f T x T x= where ( ). iT f F T , ( )1 i F T is a method defined based on the other methods and
(or) properties ( ) ( )
1
. ,..., .
nj jT x T x P T F T which form structural and (or) functional atoms, and are parts of smaller
molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of objects
T , while ( )F T is its signature.
To analyze the structure and behavior of the class Tr , we can observe that methods 1.Tr f and 2.Tr f , which get and set
the coordinates of vertices of a triangle, operate by a particular vertex of the figure. Thus, they define functional
molecules ( )1FM Tr , and ( )2FM Tr of the class Tr , i.e.
( ) ( ) ( ) ( )1 1 1 2 3 2 2 1 2 3. , . , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p FM Tr Tr f Tr p Tr p Tr p= =
In addition, method 3.Tr f , which computes the length of a particular side of a triangle, uses a corresponding pair of its
vertices. Therefore, it determines a functional molecule ( )3FM Tr of the class Tr , i.e.
( ) ( )3 3 1 2 1 3 2 3. , . , . , . , . , . , . .FM Tr Tr f Tr p Tr p Tr p Tr p Tr p Tr p=
And finally, method
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, which calculates the perimeter of a triangle, uses methods
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
and
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
to
compute the length of each figure’s side. As the result, it defines a complex structural molecule
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
of the class
Tr, which includes the elements of smaller molecules
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
and
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, i.e.
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
is a property defined based on the other
properties and (or) methods
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, which form structural and (or) functional atoms, and are
parts of smaller molecules of the class T, where
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, and
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
is a specification of the
class of objects T, while F(T) is its signature.
The class
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
has qualitative property
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, which means the satisfiability of the triangle inequality, and
uses methods
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
to compute the length of each figure’s side. Hence, it determines a complex structural
molecule
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
of the class
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, which includes the elements of smaller molecules
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, i.e.
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
Since all molecules contain a property or method which is dependent on all other properties and (or) methods
of the molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
of a homogeneous class
of objects Т is a property or method
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
which is defined based on other properties and (or) methods of the class T,
which are atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain
if such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T, which defines type of
objects t, is a set of structural and functional atoms and molecules of the class T, i.e.
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
where
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
are structural and functional atoms of the class T, while
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
,
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be
represented in the following way:
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
Let us construct the concept lattice of all internal semantic dependencies of the class Tr. For this purpose,
let us define the formal context
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
using the corresponding
cross table. Since, each of the functional molecules
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
, and
Моделі і засоби систем баз даних і знань
And finally, method 4.Tr f , which calculates the perimeter of a triangle, uses methods 3.Tr f and 1.Tr f to compute the
length of each figure’s side. As the result, it defines a complex structural molecule ( )4FM Tr of the class Tr , which
includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .FM Tr Tr f Tr f Tr f Tr p Tr p Tr p=
Definition 11. A structural molecule of a homogeneous class of objects T is a collection
( ) ( )1
. , . ,..., .
ni i j jSM T T p T x T x= where ( ). iT p P T , ( )1 i P T is a property defined based on the other properties
and (or) methods ( ) ( )
1
. ,..., .
nj jT x T x P T F T , which form structural and (or) functional atoms, and are parts of
smaller molecules of the class T , where ( ) ( )11 ... nj j P T F T , and ( )P T is a specification of the class of
objects T , while ( )F T is its signature.
The class Tr has qualitative property 4.Tr p , which means the satisfiability of the triangle inequality, and uses methods
3.Tr f and 1.Tr f to compute the length of each figure’s side. Hence, it determines a complex structural molecule 4SM
of the class Tr , which includes the elements of smaller molecules ( )1FM Tr and ( )3FM Tr , i.e.
( ) ( )4 4 3 1 1 2 3. , . , . , . , . , . .SM Tr Tr p Tr f Tr f Tr p Tr p Tr p=
Since all molecules contain a property or method which is dependent on all other properties and (or) methods of the
molecule, we can define a concept of dependency root, which describes such elements.
Definition 12. The dependency root of the molecule ( ) ( )1
. , . ,..., .
ni i j jM T T a T x T x= of a homogeneous class of objects
T is a property or method . iT a which is defined based on other properties and (or) methods of the class T , which are
atoms or parts of smaller molecules.
All detected internal dependencies within the class Tr describe some semantic connections among the different
properties and methods of the class, which express the internal nature of the modeled entity from a particular domain if
such a model is correct.
Definition 13. Internal semantic dependencies of a homogeneous class of objects T , which defines type of objects t , is
a set of structural and functional atoms and molecules of the class T , i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1,..., , ,..., , ,..., , ,..., ,n m v qISD T SA T SA T FA T FA T SM T SM T FM T FM T=
where ( )
1i
SA T , 1 1,i n= and ( )
1j
FA T , 1 1,j m= are structural and functional atoms of the class T , while ( )
2i
SM T ,
2 1,i v= and ( )
2j
FM T , 2 1,j q= are its structural and functional molecules respectively.
All considered atoms and molecules of the class Tr define its internal semantic dependencies, which can be represented
in the following way:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4, , , , , , , .ISD Tr SA Tr SA Tr SA Tr SM Tr FM Tr FM Tr FM Tr FM Tr=
Let us construct the concept lattice of all internal semantic dependencies of the class Tr . For this purpose, let us define
the formal context ( ) ( ) ( )( )2 , , :W G ISD Tr M P Tr F Tr I G M= = = using the corresponding cross table. Since,
each of the functional molecules ( )1FM Tr , ( )2FM Tr , and ( )3FM Tr has three different contexts within the class Tr ,
we let us split them onto separate conditions. We colored cells of Table 2, which means dependency roots of molecules
of the class Tr , using the gray color. Now let us construct the concept lattice for the formal context 2W , using Table 2.
Table 2. Formal context, which defines internal semantic dependencies of the class Tr .
2W
Properties and methods
1.Tr p 2.Tr p 3.Tr p 4.Tr p 1.Tr f 2.Tr f 3.Tr f 4.Tr f
A
to
m
s a
nd
m
ol
e
cu
le
s ( )1SA Tr +
( )2SA Tr +
has three different
contexts within the class Tr, we let us split them onto separate conditions. We colored cells of Table 2, which
means dependency roots of molecules of the class Tr, using the gray color. Now let us construct the concept lattice
for the formal context W2, using Table 2.
145
Моделі і засоби систем баз даних та знань
Table 2. Formal context, which defines internal semantic dependencies of the class Tr.
W2
Properties and methods
Tr.p1 Tr.p2 Tr.p3 Tr.p4 Tr.f1 Tr.f2 Tr.f3 Tr.f4
A
to
m
s a
nd
m
ol
ec
ul
es
SA1 (Tr) +
SA2 (Tr) +
SA3 (Tr) +
SM4 (Tr) + + + + + + +
FM11 (Tr) + +
FM12 (Tr) + +
FM13 (Tr) + +
FM21 (Tr) + +
FM22 (Tr) + +
FM23 (Tr) + +
FM31 (Tr) + + +
FM23 (Tr) + + +
FM33 (Tr) + + +
FM4 (Tr) + + + + + +
Analyzing the results, depicted in Figure 2, we can see that constructed concept lattice contains 22 nodes,
which means formal concepts, however not all of them are semantically consistent ones. We used the green border to
highlight the consistent concepts, which do not contradict any internal semantic dependency of the class Tr. Indeed, if
we consider, for example, the concept 14, which defined as follows
Моделі і засоби систем баз даних і знань
[Введите текст]
( )3SA Tr +
( )4SM Tr + + + + + + +
( )11FM Tr + +
( )12FM Tr + +
( )13FM Tr + +
( )21FM Tr + +
( )22FM Tr + +
( )23FM Tr + +
( )31FM Tr + + +
( )23FM Tr + + +
( )33FM Tr + + +
( )4FM Tr + + + + + +
Analyzing the results, depicted in Figure 2, we can see that constructed concept lattice contains 22 nodes, which means
formal concepts, however not all of them are semantically consistent ones. We used the green border to highlight the
consistent concepts, which do not contradict any internal semantic dependency of the class Tr . Indeed, if we consider,
for example, the concept 14 , which defined as follows
( ) ( ) ( ) ( ) ( )3 1 13 21 23 31. , . , , , , ,I Tr f Tr p E FM Tr FM Tr FM Tr FM Tr= =
we can see that the intent 3 1. , .I Tr f Tr p= of the concept is semantically inconsistent because it contradicts the
internal semantic dependencies defined by functional molecules ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr .
In other words, the concept 14 defines a point on a plane, however, it has a method 3.Tr f , which computes the distance
between two points on a plane, but the concept defines only one point, that means if we invoke this method, it will raise
an error.
Summarizing all noted above, we can conclude that classical methods of formal concept analysis have some gaps
related to the semantic consistency of formal concepts, which are constructed within a formal context of decomposition
of homogeneous classes of objects. Since, the definition of a formal context does not consider the internal semantic
dependencies of a class, defined by its atoms and molecules, the knowledge retrieval or reasoning within a
corresponding concept lattice, constructed using such formal context, becomes inconsistent too because its result can
contain inconsistent concepts.
Decomposition of Classes
Since a homogeneous class of objects consist of structural and functional molecules, which define the
restrictions over the class specification and signature, the decomposition of the class can be considered as the constraint
Figure 2. Fully annotated concept lattice of the formal context 2W .
we can see that the intent
Моделі і засоби систем баз даних і знань
[Введите текст]
( )3SA Tr +
( )4SM Tr + + + + + + +
( )11FM Tr + +
( )12FM Tr + +
( )13FM Tr + +
( )21FM Tr + +
( )22FM Tr + +
( )23FM Tr + +
( )31FM Tr + + +
( )23FM Tr + + +
( )33FM Tr + + +
( )4FM Tr + + + + + +
Analyzing the results, depicted in Figure 2, we can see that constructed concept lattice contains 22 nodes, which means
formal concepts, however not all of them are semantically consistent ones. We used the green border to highlight the
consistent concepts, which do not contradict any internal semantic dependency of the class Tr . Indeed, if we consider,
for example, the concept 14 , which defined as follows
( ) ( ) ( ) ( ) ( )3 1 13 21 23 31. , . , , , , ,I Tr f Tr p E FM Tr FM Tr FM Tr FM Tr= =
we can see that the intent 3 1. , .I Tr f Tr p= of the concept is semantically inconsistent because it contradicts the
internal semantic dependencies defined by functional molecules ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr .
In other words, the concept 14 defines a point on a plane, however, it has a method 3.Tr f , which computes the distance
between two points on a plane, but the concept defines only one point, that means if we invoke this method, it will raise
an error.
Summarizing all noted above, we can conclude that classical methods of formal concept analysis have some gaps
related to the semantic consistency of formal concepts, which are constructed within a formal context of decomposition
of homogeneous classes of objects. Since, the definition of a formal context does not consider the internal semantic
dependencies of a class, defined by its atoms and molecules, the knowledge retrieval or reasoning within a
corresponding concept lattice, constructed using such formal context, becomes inconsistent too because its result can
contain inconsistent concepts.
Decomposition of Classes
Since a homogeneous class of objects consist of structural and functional molecules, which define the
restrictions over the class specification and signature, the decomposition of the class can be considered as the constraint
Figure 2. Fully annotated concept lattice of the formal context 2W .
of the concept is semantically inconsistent because it contradicts the
internal semantic dependencies defined by functional molecules
Моделі і засоби систем баз даних і знань
[Введите текст]
( )3SA Tr +
( )4SM Tr + + + + + + +
( )11FM Tr + +
( )12FM Tr + +
( )13FM Tr + +
( )21FM Tr + +
( )22FM Tr + +
( )23FM Tr + +
( )31FM Tr + + +
( )23FM Tr + + +
( )33FM Tr + + +
( )4FM Tr + + + + + +
Analyzing the results, depicted in Figure 2, we can see that constructed concept lattice contains 22 nodes, which means
formal concepts, however not all of them are semantically consistent ones. We used the green border to highlight the
consistent concepts, which do not contradict any internal semantic dependency of the class Tr . Indeed, if we consider,
for example, the concept 14 , which defined as follows
( ) ( ) ( ) ( ) ( )3 1 13 21 23 31. , . , , , , ,I Tr f Tr p E FM Tr FM Tr FM Tr FM Tr= =
we can see that the intent 3 1. , .I Tr f Tr p= of the concept is semantically inconsistent because it contradicts the
internal semantic dependencies defined by functional molecules ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr .
In other words, the concept 14 defines a point on a plane, however, it has a method 3.Tr f , which computes the distance
between two points on a plane, but the concept defines only one point, that means if we invoke this method, it will raise
an error.
Summarizing all noted above, we can conclude that classical methods of formal concept analysis have some gaps
related to the semantic consistency of formal concepts, which are constructed within a formal context of decomposition
of homogeneous classes of objects. Since, the definition of a formal context does not consider the internal semantic
dependencies of a class, defined by its atoms and molecules, the knowledge retrieval or reasoning within a
corresponding concept lattice, constructed using such formal context, becomes inconsistent too because its result can
contain inconsistent concepts.
Decomposition of Classes
Since a homogeneous class of objects consist of structural and functional molecules, which define the
restrictions over the class specification and signature, the decomposition of the class can be considered as the constraint
Figure 2. Fully annotated concept lattice of the formal context 2W .
, and
Моделі і засоби систем баз даних і знань
[Введите текст]
( )3SA Tr +
( )4SM Tr + + + + + + +
( )11FM Tr + +
( )12FM Tr + +
( )13FM Tr + +
( )21FM Tr + +
( )22FM Tr + +
( )23FM Tr + +
( )31FM Tr + + +
( )23FM Tr + + +
( )33FM Tr + + +
( )4FM Tr + + + + + +
Analyzing the results, depicted in Figure 2, we can see that constructed concept lattice contains 22 nodes, which means
formal concepts, however not all of them are semantically consistent ones. We used the green border to highlight the
consistent concepts, which do not contradict any internal semantic dependency of the class Tr . Indeed, if we consider,
for example, the concept 14 , which defined as follows
( ) ( ) ( ) ( ) ( )3 1 13 21 23 31. , . , , , , ,I Tr f Tr p E FM Tr FM Tr FM Tr FM Tr= =
we can see that the intent 3 1. , .I Tr f Tr p= of the concept is semantically inconsistent because it contradicts the
internal semantic dependencies defined by functional molecules ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr .
In other words, the concept 14 defines a point on a plane, however, it has a method 3.Tr f , which computes the distance
between two points on a plane, but the concept defines only one point, that means if we invoke this method, it will raise
an error.
Summarizing all noted above, we can conclude that classical methods of formal concept analysis have some gaps
related to the semantic consistency of formal concepts, which are constructed within a formal context of decomposition
of homogeneous classes of objects. Since, the definition of a formal context does not consider the internal semantic
dependencies of a class, defined by its atoms and molecules, the knowledge retrieval or reasoning within a
corresponding concept lattice, constructed using such formal context, becomes inconsistent too because its result can
contain inconsistent concepts.
Decomposition of Classes
Since a homogeneous class of objects consist of structural and functional molecules, which define the
restrictions over the class specification and signature, the decomposition of the class can be considered as the constraint
Figure 2. Fully annotated concept lattice of the formal context 2W .
.
In other words, the concept 14 defines a point on a plane, however, it has a method
Моделі і засоби систем баз даних і знань
[Введите текст]
( )3SA Tr +
( )4SM Tr + + + + + + +
( )11FM Tr + +
( )12FM Tr + +
( )13FM Tr + +
( )21FM Tr + +
( )22FM Tr + +
( )23FM Tr + +
( )31FM Tr + + +
( )23FM Tr + + +
( )33FM Tr + + +
( )4FM Tr + + + + + +
Analyzing the results, depicted in Figure 2, we can see that constructed concept lattice contains 22 nodes, which means
formal concepts, however not all of them are semantically consistent ones. We used the green border to highlight the
consistent concepts, which do not contradict any internal semantic dependency of the class Tr . Indeed, if we consider,
for example, the concept 14 , which defined as follows
( ) ( ) ( ) ( ) ( )3 1 13 21 23 31. , . , , , , ,I Tr f Tr p E FM Tr FM Tr FM Tr FM Tr= =
we can see that the intent 3 1. , .I Tr f Tr p= of the concept is semantically inconsistent because it contradicts the
internal semantic dependencies defined by functional molecules ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr .
In other words, the concept 14 defines a point on a plane, however, it has a method 3.Tr f , which computes the distance
between two points on a plane, but the concept defines only one point, that means if we invoke this method, it will raise
an error.
Summarizing all noted above, we can conclude that classical methods of formal concept analysis have some gaps
related to the semantic consistency of formal concepts, which are constructed within a formal context of decomposition
of homogeneous classes of objects. Since, the definition of a formal context does not consider the internal semantic
dependencies of a class, defined by its atoms and molecules, the knowledge retrieval or reasoning within a
corresponding concept lattice, constructed using such formal context, becomes inconsistent too because its result can
contain inconsistent concepts.
Decomposition of Classes
Since a homogeneous class of objects consist of structural and functional molecules, which define the
restrictions over the class specification and signature, the decomposition of the class can be considered as the constraint
Figure 2. Fully annotated concept lattice of the formal context 2W .
, which computes the
distance between two points on a plane, but the concept defines only one point, that means if we invoke this method,
it will raise an error.
Моделі і засоби систем баз даних і знань
[Введите текст]
( )3SA Tr +
( )4SM Tr + + + + + + +
( )11FM Tr + +
( )12FM Tr + +
( )13FM Tr + +
( )21FM Tr + +
( )22FM Tr + +
( )23FM Tr + +
( )31FM Tr + + +
( )23FM Tr + + +
( )33FM Tr + + +
( )4FM Tr + + + + + +
Analyzing the results, depicted in Figure 2, we can see that constructed concept lattice contains 22 nodes, which means
formal concepts, however not all of them are semantically consistent ones. We used the green border to highlight the
consistent concepts, which do not contradict any internal semantic dependency of the class Tr . Indeed, if we consider,
for example, the concept 14 , which defined as follows
( ) ( ) ( ) ( ) ( )3 1 13 21 23 31. , . , , , , ,I Tr f Tr p E FM Tr FM Tr FM Tr FM Tr= =
we can see that the intent 3 1. , .I Tr f Tr p= of the concept is semantically inconsistent because it contradicts the
internal semantic dependencies defined by functional molecules ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr .
In other words, the concept 14 defines a point on a plane, however, it has a method 3.Tr f , which computes the distance
between two points on a plane, but the concept defines only one point, that means if we invoke this method, it will raise
an error.
Summarizing all noted above, we can conclude that classical methods of formal concept analysis have some gaps
related to the semantic consistency of formal concepts, which are constructed within a formal context of decomposition
of homogeneous classes of objects. Since, the definition of a formal context does not consider the internal semantic
dependencies of a class, defined by its atoms and molecules, the knowledge retrieval or reasoning within a
corresponding concept lattice, constructed using such formal context, becomes inconsistent too because its result can
contain inconsistent concepts.
Decomposition of Classes
Since a homogeneous class of objects consist of structural and functional molecules, which define the
restrictions over the class specification and signature, the decomposition of the class can be considered as the constraint
Figure 2. Fully annotated concept lattice of the formal context 2W . Figure 2. Fully annotated concept lattice of the formal context W2.
Summarizing all noted above, we can conclude that classical methods of formal concept analysis have
some gaps related to the semantic consistency of formal concepts, which are constructed within a formal context
146
Моделі і засоби систем баз даних та знань
of decomposition of homogeneous classes of objects. Since, the definition of a formal context does not consider the
internal semantic dependencies of a class, defined by its atoms and molecules, the knowledge retrieval or reasoning
within a corresponding concept lattice, constructed using such formal context, becomes inconsistent too because its
result can contain inconsistent concepts.
Decomposition of Classes
Since a homogeneous class of objects consist of structural and functional molecules, which define the
restrictions over the class specification and signature, the decomposition of the class can be considered as the
constraint satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, where
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is a finite sequence of variables,
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is a set of domains, and
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is a finite set of constraints. Variables from the set X are defined on domains from the set D, i.e.
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, and each domain defines a range of values for the respective variable, i.e.
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
. Every
constraint from the set C is defined as a pair
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is a scope of
the constraint and
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is a relation defined over the
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, and
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is the arity of the
constraint
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
. The tuple
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
satisfies the constraint
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
on the variables
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
if
and only if
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
. If the tuple
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
satisfies
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, then it is a solution of the CSP.
However, a particular subclass
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
of a homogeneous class of objects T can satisfy or not satisfy
a constraint defined by a molecule
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
. Therefore, in contrast to the classical definition of the CSP, the
constraint defined by the molecule
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is applicable only to some subclasses of the class T. For example, the
constraint defined by functional molecule
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is not applicable to any subclass of the class Tr, which have
a cardinality lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict
the constraint defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let
us introduce the following definition.
Definition 14. A subclass Ti of a homogeneous class of objects T does not contradict molecular internal
semantic dependency
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e.
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
;
2. it does not contain any element of the molecule, i.e.
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
where
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
is a property or method defined based on the other properties and
(or) methods
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
of the class T, where
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, and
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
and
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
are specifications and signatures of the class T and Ti respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T, which defines a type of objects t, is a
set of semantically consistent subclasses
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, where subclasses
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
do not contradict any
molecular internal semantic dependency of the class T.
Now, let us compute the full decomposition of the homogeneous class of objects Tr, using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
, which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr, where three subclasses of the cardinality of 1, i.e.
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e.
nine subclasses of the cardinality of 2, i.e.
Моделі і засоби систем баз даних і знань
satisfaction problem (CSP) [28]. According to [3, 12, 19], the CSP can be defined as a tuple ( ), ,X D C , where
1,..., nX x x= , 0n is a finite sequence of variables, 1,..., nD d d= is a set of domains, and 1,..., mC c c= , 0m
is a finite set of constraints. Variables from the set X are defined on domains from the set D , i.e. 1 1,..., n nx d x d→ → ,
and each domain defines a range of values for the respective variable, i.e. 1,...,i kd v v= , 1,i n= , 0k . Every
constraint from the set C is defined as a pair ( ),j j jc S R= , 1,j m= , where ( )1 1
,...,
w wj j j j jS x d x d= → → is a scope of
the constraint and
1
...
wj j jR d d is a relation defined over the jS , 10 ... wj j n , and 0 w n is the arity of
the constraint jc . The tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies the constraint ( ),j j jc S R= on the variables
1,..., wx x if and only if ( )1
,...,
wi i jy y S . If the tuple ( )1 1 1: ,..., :
wi i w wy x d y x d→ → satisfies jc C , then it is a
solution of the CSP.
However, a particular subclass iT T of a homogeneous class of objects T can satisfy or not satisfy a
constraint defined by a molecule ( )jM T . Therefore, in contrast to the classical definition of the CSP, the constraint
defined by the molecule ( )jM T is applicable only to some subclasses of the class T . For example, the constraint
defined by functional molecule ( )4FM Tr is not applicable to any subclass of the class Tr , which have a cardinality
lower than the molecule itself. In such cases, we can conclude that the subclass does not contradict the constraint
defined by the molecule, since the constraint is not applicable to it. To summarize these facts, let us introduce the
following definition.
Definition 14. A subclass iT of a homogeneous class of objects T does not contradict molecular internal semantic
dependency ( ) ( )1
. , . ,..., .
mj j k kM T T a T x T x= , if and only if one of the following conditions is true:
1. it contains all elements of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
2. it does not contain any element of the molecule, i.e. ( )jx M T , ( ) ( )i ix P T F T ;
3. it does not contain the dependency root of the molecule, but it contains some of its other elements, i.e.
( ) ( ). j i iT a P T F T , and 1
. ,..., .
mk kx T x T x , ( ) ( )i ix P T F T ;
where ( ) ( ). jT a P T F T , ( ) ( )1 j P T F T is a property or method defined based on the other properties and
(or) methods ( ) ( )
1
. ,..., .
mk kT x T x P T F T of the class T , where ( ) ( )11 ... mk k P T F T , and ( )P T , ( )iP T
and ( )F T , ( )iF T are specifications and signatures of the class T and iT respectively.
Using this notion, we can define the decomposition of the homogeneous classes of objects.
Definition 15. Decomposition of a homogeneous class of objects T , which defines a type of objects t , is a set of
semantically consistent subclasses ( ) 1 ,..., nD T T T T T= , where subclasses 1,..., nT T do not contradict any
molecular internal semantic dependency of the class T .
Now, let us compute the full decomposition of the homogeneous class of objects Tr , using the corresponding
algorithm, which was proposed in [28], and the set of internal semantic dependencies ( )ISD Tr , which defines a
collection of decomposition constraints. As the result of decomposition we obtained the collection of 49 semantically
consistent subclasses of the class Tr , where three subclasses of the cardinality of 1, i.e.
( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1 2 3 3 3. , . , . ,SC Tr Tr p SC Tr Tr p SC Tr Tr p= = =
nine subclasses of the cardinality of 2, i.e.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2 2 2
1 1 2 2 1 3 3 2 3 4 1 1
2 2 2 2
5 2 1 6 3 1 7 1 2 8 2 2
2
9 3 2
. , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . ,
. , . ,
SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr p SC Tr Tr p Tr f
SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f SC Tr Tr p Tr f
SC Tr Tr p Tr f
= = = =
= = = =
=
thirteen subclasses of the cardinality of 3, i.e. thirteen subclasses of the cardinality of 3, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W
twelve subclasses of the cardinality of 4, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W
six subclasses of the cardinality of 5, i.e.
147
Моделі і засоби систем баз даних та знань
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W
three subclasses of the cardinality of 6, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W
and three subclasses of the cardinality of 7, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr. For this
purpose, let us consider the formal context
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W
using the structure of
semantically consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal
context W3, depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent
subclasses constructed by the decomposition algorithm is equal to 49. It means that some formal concepts in the lattice
are semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W
,
similarly to the case of concept 14 in formal context W2. It happens because the algorithms for constructing of
concept lattices compute the part of extents as the intersection of those extents which can be extracted from the
formal context cross table [21]. They do not consider internal semantic dependencies within the classes and
objects, consequently, they escape a question about the existence of such concepts within a modeled domain,
rather compute only intersection among objects or classes to obtain sets of common attributes as new concepts.
We think that it is an important restriction for the usage of formal concept analysis, in particular, for knowledge
retrieval and reasoning, since there is an ability to retrieve or infer concepts, which are inconsistent, and therefore
unreal within a modeled domain.
Let us compare the amount of semantically consistent subclasses of the class Tr with the amount of all possible
subclasses, splitting them according to antichains of join-semilattice created by all subclasses.
Моделі і засоби систем баз даних і знань
[Введите текст]
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 3
1 1 2 3 2 1 2 1 3 1 3 1
3 3 3
4 2 3 1 5 1 2 2 6 1 3 2
3 3
7 2 3 2 8 1 1 2
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f SC Tr Tr p Tr f Tr f
= = =
= = =
= = ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
3
9 2 1 2
3 3 3
10 3 1 2 11 1 2 3 12 1 3 3
3
13 2 3 3
. , . , . ,
. , . , . , . , . , . , . , . , . ,
. , . , . ,
SC Tr Tr p Tr f Tr f
SC Tr Tr p Tr f Tr f SC Tr Tr p Tr p Tr f SC Tr Tr p Tr p Tr f
SC Tr Tr p Tr p Tr f
=
= = =
=
twelve subclasses of the cardinality of 4, i.e.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
4 4 4
1 1 2 3 1 2 1 2 3 2 3 1 2 1 2
4 4 4
4 1 3 1 2 5 2 3 1 2 6 1 2 3 3
4
7 1
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
. ,
SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr p Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f
SC Tr Tr p T
= = =
= = =
= ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4 4
2 1 3 8 1 3 1 3 9 2 3 1 3
4 4 4
10 1 2 2 3 11 1 3 2 3 12 2 3 2 3
. , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . , . , . ,
r p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f
= =
= = =
six subclasses of the cardinality of 5, i.e.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
5 5
1 1 2 3 1 2 2 1 2 3 1 3
5 5
3 1 2 3 2 3 4 1 2 1 2 3
5 5
5 1 3 1 2 3 6 2
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr
= =
= =
= = ( )3 1 2 3. , . , . , . ,p Tr f Tr f Tr f
three subclasses of the cardinality of 6, i.e.
( ) ( ) ( ) ( )
( ) ( )
6 6
1 1 2 3 4 1 3 2 1 2 3 1 2 3
6
3 1 2 3 1 3 4
. , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . ,
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f
= =
=
and three subclasses of the cardinality of 7, i.e.
( ) ( ) ( ) ( )
( ) ( )
7 7
1 1 2 3 4 1 2 3 2 1 2 3 4 1 3 4
7
3 1 2 3 1 2 3 4
. , . , . , . , . , . , . , . , . , . , . , . , . , . ,
. , . , . , . , . , . , . .
SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f SC Tr Tr p Tr p Tr p Tr p Tr f Tr f Tr f
SC Tr Tr p Tr p Tr p Tr f Tr f Tr f Tr f
= =
=
Now let us construct the concept lattice for all semantically consistent subclasses of the class Tr . For this purpose, let
us consider the formal context ( ) ( ) ( )( )3 , , :W G D Tr M P Tr F Tr I G M= = = using the structure of semantically
consistent subclasses of the class Tr noted above. As we can see, the concept lattice for the formal context 3W ,
depicted in Figure 3, contains 70 formal concepts, while the amount of all semantically consistent subclasses
constructed by the decomposition algorithm is equal to 49 . It means that some formal concepts in the lattice are
semantically consistent, while other ones are inconsistent. For example, the formal concept 44 is semantically
inconsistent because its intent contradicts internal semantic dependencies ( )31FM Tr , ( )32FM Tr , and ( )33FM Tr ,
similarly to the case of concept 14 in formal context 2W . It happens because the algorithms for constructing of concept
lattices compute the part of extents as the intersection of those extents which can be extracted from the formal context
cross table [21]. They do not consider internal semantic dependencies within the classes and objects, consequently, they
escape a question about the existence of such concepts within a modeled domain, rather compute only intersection
among objects or classes to obtain sets of common attributes as new concepts. We think that it is an important
restriction for the usage of formal concept analysis, in particular, for knowledge retrieval and reasoning, since there is
an ability to retrieve or infer concepts, which are inconsistent, and therefore unreal within a modeled domain.
Figure 3. Partially annotated concept lattice of the formal context 3W Table 3. Quantitate analysis of subclasses of the class W3.
Cardinality 1 2 3 4 5 6 7 Total
Possible subclasses 8 28 56 70 56 28 8 254
Consistent subclasses 3 9 13 12 6 3 3 49
Analyzing Table 1, we can see that among all 254 formally possible nonempty proper subclasses of the
homogeneous class of objects Tr, only 49, i.e. 19.3%, are semantically consistent ones, i.e. they do not contradict any
148
Моделі і засоби систем баз даних та знань
of the internal semantic dependencies of the class. This coefficient allows us to estimate how the search space can
be reduced by avoiding the consideration of all semantically inconsistent subclasses of the class Tr. Therefore, let us
introduce the corresponding definition for it.
Definition 16. Decomposition consistency of a homogeneous class of objects T is a coefficient DC(T) computed
in the following way
Моделі і засоби систем баз даних і знань
Let us compare the amount of semantically consistent subclasses of the class Tr with the amount of all possible
subclasses, splitting them according to antichains of join-semilattice created by all subclasses.
Table 3. Quantitate analysis of subclasses of the class Tr .
Cardinality 1 2 3 4 5 6 7 Total
Possible subclasses 8 28 56 70 56 28 8 254
Consistent subclasses 3 9 13 12 6 3 3 49
Analyzing Table 1, we can see that among all 254 formally possible nonempty proper subclasses of the homogeneous
class of objects Tr , only 49, i.e. 19.3%, are semantically consistent ones, i.e. they do not contradict any of the internal
semantic dependencies of the class. This coefficient allows us to estimate how the search space can be reduced by
avoiding the consideration of all semantically inconsistent subclasses of the class Tr . Therefore, let us introduce the
corresponding definition for it.
Definition 16. Decomposition consistency of a homogeneous class of objects T is a coefficient ( )DC T computed in
the following way
( )
( )
( )
100%
2
D T
DC T
PS T
=
−
,
where ( )D T is a set of all semantically consistent subclasses of the class T , while ( )PS T is a power set of its all
possible subclasses.
Since the ( ) 19.3%DC Tr = , it means that we can reduce the knowledge search space for the class Tr approximately by
5.2 times, i.e. 100% :19.3% 5.2 .
All data given in Table 1 can be represented graphically, that provides an opportunity to estimate the search
space for the knowledge extraction from another perspective. Figure 4 illustrates elements of the power set lattice,
where each element is a subclass of the homogeneous class of objects Tr .
Circles depicted by green color mean semantically consistent subclasses, while yellow circles with numbers mean a
particular antichain of the lattice, or in other words, a set of subclasses of the corresponding cardinality. We also can
see, that each element of this lattice, which has a cardinality bigger than 2 , and lower than the class itself, can be also
decomposed into subclasses, where some of them are semantically consistent, while others are not so. We also depicted
in Figure 4 towers of subclass lattice for particular semantically consistent subclasses of cardinality from 2 to 7 ,
which illustrates that the subclass lattice tower of the class Tr contains towers of subclass lattices. The graphical
representation of the complete lattice, illustrated in Figure 4, is not a typical or common way to the depiction of lattices,
such as the Hasse diagram, for example. However, as you can see the power set lattice of the class Tr contains 256
elements and, as was noted in [32], 12nn − connections, which makes the corresponding Hasse diagram complicated.
Instead of this, for the quantitative analysis of semantically consistent subclasses of a particular homogeneous class of
objects, we can depict only elements of the lattice’s antichains. Since the geometrical form of such representation
reminds a tower, we called it a tower of the power set lattice or a tower of subclass lattice.
Figure 4. Tower of the power set lattice of all subclasses of the class Tr .
where DC(T) is a set of all semantically consistent subclasses of the class T, while PS(T) is a power set of its all
possible subclasses.
Since the
Моделі і засоби систем баз даних і знань
Let us compare the amount of semantically consistent subclasses of the class Tr with the amount of all possible
subclasses, splitting them according to antichains of join-semilattice created by all subclasses.
Table 3. Quantitate analysis of subclasses of the class Tr .
Cardinality 1 2 3 4 5 6 7 Total
Possible subclasses 8 28 56 70 56 28 8 254
Consistent subclasses 3 9 13 12 6 3 3 49
Analyzing Table 1, we can see that among all 254 formally possible nonempty proper subclasses of the homogeneous
class of objects Tr , only 49, i.e. 19.3%, are semantically consistent ones, i.e. they do not contradict any of the internal
semantic dependencies of the class. This coefficient allows us to estimate how the search space can be reduced by
avoiding the consideration of all semantically inconsistent subclasses of the class Tr . Therefore, let us introduce the
corresponding definition for it.
Definition 16. Decomposition consistency of a homogeneous class of objects T is a coefficient ( )DC T computed in
the following way
( )
( )
( )
100%
2
D T
DC T
PS T
=
−
,
where ( )D T is a set of all semantically consistent subclasses of the class T , while ( )PS T is a power set of its all
possible subclasses.
Since the ( ) 19.3%DC Tr = , it means that we can reduce the knowledge search space for the class Tr approximately by
5.2 times, i.e. 100% :19.3% 5.2 .
All data given in Table 1 can be represented graphically, that provides an opportunity to estimate the search
space for the knowledge extraction from another perspective. Figure 4 illustrates elements of the power set lattice,
where each element is a subclass of the homogeneous class of objects Tr .
Circles depicted by green color mean semantically consistent subclasses, while yellow circles with numbers mean a
particular antichain of the lattice, or in other words, a set of subclasses of the corresponding cardinality. We also can
see, that each element of this lattice, which has a cardinality bigger than 2 , and lower than the class itself, can be also
decomposed into subclasses, where some of them are semantically consistent, while others are not so. We also depicted
in Figure 4 towers of subclass lattice for particular semantically consistent subclasses of cardinality from 2 to 7 ,
which illustrates that the subclass lattice tower of the class Tr contains towers of subclass lattices. The graphical
representation of the complete lattice, illustrated in Figure 4, is not a typical or common way to the depiction of lattices,
such as the Hasse diagram, for example. However, as you can see the power set lattice of the class Tr contains 256
elements and, as was noted in [32], 12nn − connections, which makes the corresponding Hasse diagram complicated.
Instead of this, for the quantitative analysis of semantically consistent subclasses of a particular homogeneous class of
objects, we can depict only elements of the lattice’s antichains. Since the geometrical form of such representation
reminds a tower, we called it a tower of the power set lattice or a tower of subclass lattice.
Figure 4. Tower of the power set lattice of all subclasses of the class Tr .
, it means that we can reduce the knowledge search space for the class Tr
approximately by
Моделі і засоби систем баз даних і знань
Let us compare the amount of semantically consistent subclasses of the class Tr with the amount of all possible
subclasses, splitting them according to antichains of join-semilattice created by all subclasses.
Table 3. Quantitate analysis of subclasses of the class Tr .
Cardinality 1 2 3 4 5 6 7 Total
Possible subclasses 8 28 56 70 56 28 8 254
Consistent subclasses 3 9 13 12 6 3 3 49
Analyzing Table 1, we can see that among all 254 formally possible nonempty proper subclasses of the homogeneous
class of objects Tr , only 49, i.e. 19.3%, are semantically consistent ones, i.e. they do not contradict any of the internal
semantic dependencies of the class. This coefficient allows us to estimate how the search space can be reduced by
avoiding the consideration of all semantically inconsistent subclasses of the class Tr . Therefore, let us introduce the
corresponding definition for it.
Definition 16. Decomposition consistency of a homogeneous class of objects T is a coefficient ( )DC T computed in
the following way
( )
( )
( )
100%
2
D T
DC T
PS T
=
−
,
where ( )D T is a set of all semantically consistent subclasses of the class T , while ( )PS T is a power set of its all
possible subclasses.
Since the ( ) 19.3%DC Tr = , it means that we can reduce the knowledge search space for the class Tr approximately by
5.2 times, i.e. 100% :19.3% 5.2 .
All data given in Table 1 can be represented graphically, that provides an opportunity to estimate the search
space for the knowledge extraction from another perspective. Figure 4 illustrates elements of the power set lattice,
where each element is a subclass of the homogeneous class of objects Tr .
Circles depicted by green color mean semantically consistent subclasses, while yellow circles with numbers mean a
particular antichain of the lattice, or in other words, a set of subclasses of the corresponding cardinality. We also can
see, that each element of this lattice, which has a cardinality bigger than 2 , and lower than the class itself, can be also
decomposed into subclasses, where some of them are semantically consistent, while others are not so. We also depicted
in Figure 4 towers of subclass lattice for particular semantically consistent subclasses of cardinality from 2 to 7 ,
which illustrates that the subclass lattice tower of the class Tr contains towers of subclass lattices. The graphical
representation of the complete lattice, illustrated in Figure 4, is not a typical or common way to the depiction of lattices,
such as the Hasse diagram, for example. However, as you can see the power set lattice of the class Tr contains 256
elements and, as was noted in [32], 12nn − connections, which makes the corresponding Hasse diagram complicated.
Instead of this, for the quantitative analysis of semantically consistent subclasses of a particular homogeneous class of
objects, we can depict only elements of the lattice’s antichains. Since the geometrical form of such representation
reminds a tower, we called it a tower of the power set lattice or a tower of subclass lattice.
Figure 4. Tower of the power set lattice of all subclasses of the class Tr .
All data given in Table 1 can be represented graphically, that provides an opportunity to estimate the search
space for the knowledge extraction from another perspective. Figure 4 illustrates elements of the power set lattice,
where each element is a subclass of the homogeneous class of objects Tr.
Circles depicted by green color mean semantically consistent subclasses, while yellow circles with
numbers mean a particular antichain of the lattice, or in other words, a set of subclasses of the corresponding
cardinality. We also can see, that each element of this lattice, which has a cardinality bigger than 2, and lower
than the class itself, can be also decomposed into subclasses, where some of them are semantically consistent,
while others are not so. We also depicted in Figure 4 towers of subclass lattice for particular semantically
consistent subclasses of cardinality from 2 to 7, which illustrates that the subclass lattice tower of the class
Tr contains towers of subclass lattices. The graphical representation of the complete lattice, illustrated in
Figure 4, is not a typical or common way to the depiction of lattices, such as the Hasse diagram, for example.
However, as you can see the power set lattice of the class Tr contains 256 elements and, as was noted in
[32],
Моделі і засоби систем баз даних і знань
Let us compare the amount of semantically consistent subclasses of the class Tr with the amount of all possible
subclasses, splitting them according to antichains of join-semilattice created by all subclasses.
Table 3. Quantitate analysis of subclasses of the class Tr .
Cardinality 1 2 3 4 5 6 7 Total
Possible subclasses 8 28 56 70 56 28 8 254
Consistent subclasses 3 9 13 12 6 3 3 49
Analyzing Table 1, we can see that among all 254 formally possible nonempty proper subclasses of the homogeneous
class of objects Tr , only 49, i.e. 19.3%, are semantically consistent ones, i.e. they do not contradict any of the internal
semantic dependencies of the class. This coefficient allows us to estimate how the search space can be reduced by
avoiding the consideration of all semantically inconsistent subclasses of the class Tr . Therefore, let us introduce the
corresponding definition for it.
Definition 16. Decomposition consistency of a homogeneous class of objects T is a coefficient ( )DC T computed in
the following way
( )
( )
( )
100%
2
D T
DC T
PS T
=
−
,
where ( )D T is a set of all semantically consistent subclasses of the class T , while ( )PS T is a power set of its all
possible subclasses.
Since the ( ) 19.3%DC Tr = , it means that we can reduce the knowledge search space for the class Tr approximately by
5.2 times, i.e. 100% :19.3% 5.2 .
All data given in Table 1 can be represented graphically, that provides an opportunity to estimate the search
space for the knowledge extraction from another perspective. Figure 4 illustrates elements of the power set lattice,
where each element is a subclass of the homogeneous class of objects Tr .
Circles depicted by green color mean semantically consistent subclasses, while yellow circles with numbers mean a
particular antichain of the lattice, or in other words, a set of subclasses of the corresponding cardinality. We also can
see, that each element of this lattice, which has a cardinality bigger than 2 , and lower than the class itself, can be also
decomposed into subclasses, where some of them are semantically consistent, while others are not so. We also depicted
in Figure 4 towers of subclass lattice for particular semantically consistent subclasses of cardinality from 2 to 7 ,
which illustrates that the subclass lattice tower of the class Tr contains towers of subclass lattices. The graphical
representation of the complete lattice, illustrated in Figure 4, is not a typical or common way to the depiction of lattices,
such as the Hasse diagram, for example. However, as you can see the power set lattice of the class Tr contains 256
elements and, as was noted in [32], 12nn − connections, which makes the corresponding Hasse diagram complicated.
Instead of this, for the quantitative analysis of semantically consistent subclasses of a particular homogeneous class of
objects, we can depict only elements of the lattice’s antichains. Since the geometrical form of such representation
reminds a tower, we called it a tower of the power set lattice or a tower of subclass lattice.
Figure 4. Tower of the power set lattice of all subclasses of the class Tr .
connections, which makes the corresponding Hasse diagram complicated. Instead of this, for the
quantitative analysis of semantically consistent subclasses of a particular homogeneous class of objects, we
can depict only elements of the lattice’s antichains. Since the geometrical form of such representation reminds
a tower, we called it a tower of the power set lattice or a tower of subclass lattice.
Моделі і засоби систем баз даних і знань
Let us compare the amount of semantically consistent subclasses of the class Tr with the amount of all possible
subclasses, splitting them according to antichains of join-semilattice created by all subclasses.
Table 3. Quantitate analysis of subclasses of the class Tr .
Cardinality 1 2 3 4 5 6 7 Total
Possible subclasses 8 28 56 70 56 28 8 254
Consistent subclasses 3 9 13 12 6 3 3 49
Analyzing Table 1, we can see that among all 254 formally possible nonempty proper subclasses of the homogeneous
class of objects Tr , only 49, i.e. 19.3%, are semantically consistent ones, i.e. they do not contradict any of the internal
semantic dependencies of the class. This coefficient allows us to estimate how the search space can be reduced by
avoiding the consideration of all semantically inconsistent subclasses of the class Tr . Therefore, let us introduce the
corresponding definition for it.
Definition 16. Decomposition consistency of a homogeneous class of objects T is a coefficient ( )DC T computed in
the following way
( )
( )
( )
100%
2
D T
DC T
PS T
=
−
,
where ( )D T is a set of all semantically consistent subclasses of the class T , while ( )PS T is a power set of its all
possible subclasses.
Since the ( ) 19.3%DC Tr = , it means that we can reduce the knowledge search space for the class Tr approximately by
5.2 times, i.e. 100% :19.3% 5.2 .
All data given in Table 1 can be represented graphically, that provides an opportunity to estimate the search
space for the knowledge extraction from another perspective. Figure 4 illustrates elements of the power set lattice,
where each element is a subclass of the homogeneous class of objects Tr .
Circles depicted by green color mean semantically consistent subclasses, while yellow circles with numbers mean a
particular antichain of the lattice, or in other words, a set of subclasses of the corresponding cardinality. We also can
see, that each element of this lattice, which has a cardinality bigger than 2 , and lower than the class itself, can be also
decomposed into subclasses, where some of them are semantically consistent, while others are not so. We also depicted
in Figure 4 towers of subclass lattice for particular semantically consistent subclasses of cardinality from 2 to 7 ,
which illustrates that the subclass lattice tower of the class Tr contains towers of subclass lattices. The graphical
representation of the complete lattice, illustrated in Figure 4, is not a typical or common way to the depiction of lattices,
such as the Hasse diagram, for example. However, as you can see the power set lattice of the class Tr contains 256
elements and, as was noted in [32], 12nn − connections, which makes the corresponding Hasse diagram complicated.
Instead of this, for the quantitative analysis of semantically consistent subclasses of a particular homogeneous class of
objects, we can depict only elements of the lattice’s antichains. Since the geometrical form of such representation
reminds a tower, we called it a tower of the power set lattice or a tower of subclass lattice.
Figure 4. Tower of the power set lattice of all subclasses of the class Tr . Figure 4. Tower of the power set lattice of all subclasses of the class Tr.
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can
be matched by its formal sub-context constructed using three main kinds of incidence relations
Моделі і засоби систем баз даних і знань
[Введите текст]
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can be
matched by its formal sub-context constructed using three main kinds of incidence relations | ,s sgJm m m gIm ,
| ,s sgJm m m gIm , and | , /
ss s s mgJm m m gIm m , that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
( ) ( )1,... , 1kN n n k P T F T= = − ,
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically consistent
subclasses of a class T , whose cardinality is matched with one of the list N . It allows us to further reduce the search
space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of the class T , that
we want to retrieve. In addition, we can add parameter
1 1
. ,..., . , . ,..., .
w qa i i j jQ include T a T a exclude T a T a = = = ,
which means the attribute query and allows the algorithm to construct only those semantically consistent subclasses of
the class T , whose contain and do not contain properties and (or) methods from the include and exclude list
respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know useful information
about them, i.e. which properties and (or) methods they should and should not contain. Finally, we can add parameter
( ) ( ) ( ) ( )
1 1
,..., , ,...,
v md i i j jQ include d T d T exclude d T d T = = = ,
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T , whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to reduce
the number of constructed subclasses, if we know other useful information about them, i.e. elements of which structural
and (or) functional molecules they should and should not contain. Parameters aQ and dQ are filters, which allow us to
retrieve semantically consistent subclasses of the class T according to particular structural and behavior features. Using
all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous classes of objects
in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Require: , , , ,a dT C N Q Q
Ensure: D
1: : {};D =
2: for n N do
3: : {};t =
4: for 1,..., 2 1ni = − do
5: if ( ) ( )binary .count 1i i= then
6: for , 1,...,| |ja T j T = do
7: if ( )( )& 1 0i j then
8: ( ).add ;jt a
9: satisfy := true;
10: for all c C do
11: if not ( )is_satisfy ,t c then
12: satisfy := false;
13: break;
14: if satisfy then
15: if ( )satisfy_query , at Q and ( )satisfy_query , dt Q then
,
Моделі і засоби систем баз даних і знань
[Введите текст]
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can be
matched by its formal sub-context constructed using three main kinds of incidence relations | ,s sgJm m m gIm ,
| ,s sgJm m m gIm , and | , /
ss s s mgJm m m gIm m , that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
( ) ( )1,... , 1kN n n k P T F T= = − ,
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically consistent
subclasses of a class T , whose cardinality is matched with one of the list N . It allows us to further reduce the search
space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of the class T , that
we want to retrieve. In addition, we can add parameter
1 1
. ,..., . , . ,..., .
w qa i i j jQ include T a T a exclude T a T a = = = ,
which means the attribute query and allows the algorithm to construct only those semantically consistent subclasses of
the class T , whose contain and do not contain properties and (or) methods from the include and exclude list
respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know useful information
about them, i.e. which properties and (or) methods they should and should not contain. Finally, we can add parameter
( ) ( ) ( ) ( )
1 1
,..., , ,...,
v md i i j jQ include d T d T exclude d T d T = = = ,
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T , whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to reduce
the number of constructed subclasses, if we know other useful information about them, i.e. elements of which structural
and (or) functional molecules they should and should not contain. Parameters aQ and dQ are filters, which allow us to
retrieve semantically consistent subclasses of the class T according to particular structural and behavior features. Using
all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous classes of objects
in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Require: , , , ,a dT C N Q Q
Ensure: D
1: : {};D =
2: for n N do
3: : {};t =
4: for 1,..., 2 1ni = − do
5: if ( ) ( )binary .count 1i i= then
6: for , 1,...,| |ja T j T = do
7: if ( )( )& 1 0i j then
8: ( ).add ;jt a
9: satisfy := true;
10: for all c C do
11: if not ( )is_satisfy ,t c then
12: satisfy := false;
13: break;
14: if satisfy then
15: if ( )satisfy_query , at Q and ( )satisfy_query , dt Q then
, that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
Моделі і засоби систем баз даних і знань
[Введите текст]
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can be
matched by its formal sub-context constructed using three main kinds of incidence relations | ,s sgJm m m gIm ,
| ,s sgJm m m gIm , and | , /
ss s s mgJm m m gIm m , that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
( ) ( )1,... , 1kN n n k P T F T= = − ,
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically consistent
subclasses of a class T , whose cardinality is matched with one of the list N . It allows us to further reduce the search
space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of the class T , that
we want to retrieve. In addition, we can add parameter
1 1
. ,..., . , . ,..., .
w qa i i j jQ include T a T a exclude T a T a = = = ,
which means the attribute query and allows the algorithm to construct only those semantically consistent subclasses of
the class T , whose contain and do not contain properties and (or) methods from the include and exclude list
respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know useful information
about them, i.e. which properties and (or) methods they should and should not contain. Finally, we can add parameter
( ) ( ) ( ) ( )
1 1
,..., , ,...,
v md i i j jQ include d T d T exclude d T d T = = = ,
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T , whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to reduce
the number of constructed subclasses, if we know other useful information about them, i.e. elements of which structural
and (or) functional molecules they should and should not contain. Parameters aQ and dQ are filters, which allow us to
retrieve semantically consistent subclasses of the class T according to particular structural and behavior features. Using
all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous classes of objects
in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Require: , , , ,a dT C N Q Q
Ensure: D
1: : {};D =
2: for n N do
3: : {};t =
4: for 1,..., 2 1ni = − do
5: if ( ) ( )binary .count 1i i= then
6: for , 1,...,| |ja T j T = do
7: if ( )( )& 1 0i j then
8: ( ).add ;jt a
9: satisfy := true;
10: for all c C do
11: if not ( )is_satisfy ,t c then
12: satisfy := false;
13: break;
14: if satisfy then
15: if ( )satisfy_query , at Q and ( )satisfy_query , dt Q then
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically
consistent subclasses of a class T, whose cardinality is matched with one of the list N. It allows us to further reduce
149
Моделі і засоби систем баз даних та знань
the search space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of
the class T, that we want to retrieve. In addition, we can add parameter
Моделі і засоби систем баз даних і знань
[Введите текст]
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can be
matched by its formal sub-context constructed using three main kinds of incidence relations | ,s sgJm m m gIm ,
| ,s sgJm m m gIm , and | , /
ss s s mgJm m m gIm m , that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
( ) ( )1,... , 1kN n n k P T F T= = − ,
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically consistent
subclasses of a class T , whose cardinality is matched with one of the list N . It allows us to further reduce the search
space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of the class T , that
we want to retrieve. In addition, we can add parameter
1 1
. ,..., . , . ,..., .
w qa i i j jQ include T a T a exclude T a T a = = = ,
which means the attribute query and allows the algorithm to construct only those semantically consistent subclasses of
the class T , whose contain and do not contain properties and (or) methods from the include and exclude list
respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know useful information
about them, i.e. which properties and (or) methods they should and should not contain. Finally, we can add parameter
( ) ( ) ( ) ( )
1 1
,..., , ,...,
v md i i j jQ include d T d T exclude d T d T = = = ,
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T , whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to reduce
the number of constructed subclasses, if we know other useful information about them, i.e. elements of which structural
and (or) functional molecules they should and should not contain. Parameters aQ and dQ are filters, which allow us to
retrieve semantically consistent subclasses of the class T according to particular structural and behavior features. Using
all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous classes of objects
in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Require: , , , ,a dT C N Q Q
Ensure: D
1: : {};D =
2: for n N do
3: : {};t =
4: for 1,..., 2 1ni = − do
5: if ( ) ( )binary .count 1i i= then
6: for , 1,...,| |ja T j T = do
7: if ( )( )& 1 0i j then
8: ( ).add ;jt a
9: satisfy := true;
10: for all c C do
11: if not ( )is_satisfy ,t c then
12: satisfy := false;
13: break;
14: if satisfy then
15: if ( )satisfy_query , at Q and ( )satisfy_query , dt Q then
which means the attribute query and allows the algorithm to construct only those semantically consistent
subclasses of the class T, whose contain and do not contain properties and (or) methods from the include and
exclude list respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know
useful information about them, i.e. which properties and (or) methods they should and should not contain. Finally,
we can add parameter
Моделі і засоби систем баз даних і знань
[Введите текст]
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can be
matched by its formal sub-context constructed using three main kinds of incidence relations | ,s sgJm m m gIm ,
| ,s sgJm m m gIm , and | , /
ss s s mgJm m m gIm m , that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
( ) ( )1,... , 1kN n n k P T F T= = − ,
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically consistent
subclasses of a class T , whose cardinality is matched with one of the list N . It allows us to further reduce the search
space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of the class T , that
we want to retrieve. In addition, we can add parameter
1 1
. ,..., . , . ,..., .
w qa i i j jQ include T a T a exclude T a T a = = = ,
which means the attribute query and allows the algorithm to construct only those semantically consistent subclasses of
the class T , whose contain and do not contain properties and (or) methods from the include and exclude list
respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know useful information
about them, i.e. which properties and (or) methods they should and should not contain. Finally, we can add parameter
( ) ( ) ( ) ( )
1 1
,..., , ,...,
v md i i j jQ include d T d T exclude d T d T = = = ,
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T , whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to reduce
the number of constructed subclasses, if we know other useful information about them, i.e. elements of which structural
and (or) functional molecules they should and should not contain. Parameters aQ and dQ are filters, which allow us to
retrieve semantically consistent subclasses of the class T according to particular structural and behavior features. Using
all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous classes of objects
in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Require: , , , ,a dT C N Q Q
Ensure: D
1: : {};D =
2: for n N do
3: : {};t =
4: for 1,..., 2 1ni = − do
5: if ( ) ( )binary .count 1i i= then
6: for , 1,...,| |ja T j T = do
7: if ( )( )& 1 0i j then
8: ( ).add ;jt a
9: satisfy := true;
10: for all c C do
11: if not ( )is_satisfy ,t c then
12: satisfy := false;
13: break;
14: if satisfy then
15: if ( )satisfy_query , at Q and ( )satisfy_query , dt Q then
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T, whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to
reduce the number of constructed subclasses, if we know other useful information about them, i.e. elements of which
structural and (or) functional molecules they should and should not contain. Parameters
Моделі і засоби систем баз даних і знань
[Введите текст]
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can be
matched by its formal sub-context constructed using three main kinds of incidence relations | ,s sgJm m m gIm ,
| ,s sgJm m m gIm , and | , /
ss s s mgJm m m gIm m , that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
( ) ( )1,... , 1kN n n k P T F T= = − ,
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically consistent
subclasses of a class T , whose cardinality is matched with one of the list N . It allows us to further reduce the search
space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of the class T , that
we want to retrieve. In addition, we can add parameter
1 1
. ,..., . , . ,..., .
w qa i i j jQ include T a T a exclude T a T a = = = ,
which means the attribute query and allows the algorithm to construct only those semantically consistent subclasses of
the class T , whose contain and do not contain properties and (or) methods from the include and exclude list
respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know useful information
about them, i.e. which properties and (or) methods they should and should not contain. Finally, we can add parameter
( ) ( ) ( ) ( )
1 1
,..., , ,...,
v md i i j jQ include d T d T exclude d T d T = = = ,
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T , whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to reduce
the number of constructed subclasses, if we know other useful information about them, i.e. elements of which structural
and (or) functional molecules they should and should not contain. Parameters aQ and dQ are filters, which allow us to
retrieve semantically consistent subclasses of the class T according to particular structural and behavior features. Using
all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous classes of objects
in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Require: , , , ,a dT C N Q Q
Ensure: D
1: : {};D =
2: for n N do
3: : {};t =
4: for 1,..., 2 1ni = − do
5: if ( ) ( )binary .count 1i i= then
6: for , 1,...,| |ja T j T = do
7: if ( )( )& 1 0i j then
8: ( ).add ;jt a
9: satisfy := true;
10: for all c C do
11: if not ( )is_satisfy ,t c then
12: satisfy := false;
13: break;
14: if satisfy then
15: if ( )satisfy_query , at Q and ( )satisfy_query , dt Q then
are filters, which
allow us to retrieve semantically consistent subclasses of the class T according to particular structural and behavior
features. Using all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous
classes of objects in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Моделі і засоби систем баз даних і знань
[Введите текст]
Knowledge Extraction and Retrieval
One of the approaches to knowledge retrieval was proposed in [24], according to which a formal context can be
matched by its formal sub-context constructed using three main kinds of incidence relations | ,s sgJm m m gIm ,
| ,s sgJm m m gIm , and | , /
ss s s mgJm m m gIm m , that allows the cauterization of the formal context.
However, such an approach requires constructing additional concept lattices, to perform their matching with the main
concept lattice creating clusters, that can affect the performance of knowledge extraction. Therefore, let us consider
another approach.
As we can see, the algorithm for decomposition of homogeneous classes of objects, proposed in [28], can be
used for knowledge extraction of semantically consistent subclasses of a class. However, it also can be adapted for
knowledge retrieval, by adding additional filtration parameters, which will provide new functional opportunities for
conceptual knowledge retrieval and speed up the retrieval process itself. For this purpose, let us add the parameter
( ) ( )1,... , 1kN n n k P T F T= = − ,
which means the list of subclass cardinalities and allows the algorithm to construct only those semantically consistent
subclasses of a class T , whose cardinality is matched with one of the list N . It allows us to further reduce the search
space for the algorithm if we know the exact cardinality of the semantically consistent subclasses of the class T , that
we want to retrieve. In addition, we can add parameter
1 1
. ,..., . , . ,..., .
w qa i i j jQ include T a T a exclude T a T a = = = ,
which means the attribute query and allows the algorithm to construct only those semantically consistent subclasses of
the class T , whose contain and do not contain properties and (or) methods from the include and exclude list
respectively. It also helps the algorithm to reduce the number of constructed subclasses, if we know useful information
about them, i.e. which properties and (or) methods they should and should not contain. Finally, we can add parameter
( ) ( ) ( ) ( )
1 1
,..., , ,...,
v md i i j jQ include d T d T exclude d T d T = = = ,
which means the dependency query and allows the algorithm to construct only those semantically consistent subclasses
of the class T , whose contain and do not contain properties and (or) methods that are parts of internal semantic
dependencies, from include and exclude lists respectively. Similar to the attributes, it also helps the algorithm to reduce
the number of constructed subclasses, if we know other useful information about them, i.e. elements of which structural
and (or) functional molecules they should and should not contain. Parameters aQ and dQ are filters, which allow us to
retrieve semantically consistent subclasses of the class T according to particular structural and behavior features. Using
all these filtration parameters, we can improve the algorithm for the decomposition of homogeneous classes of objects
in the following way.
Algorithm 1. Decomposition of homogeneous class of objects.
Require: , , , ,a dT C N Q Q
Ensure: D
1: : {};D =
2: for n N do
3: : {};t =
4: for 1,..., 2 1ni = − do
5: if ( ) ( )binary .count 1i i= then
6: for , 1,...,| |ja T j T = do
7: if ( )( )& 1 0i j then
8: ( ).add ;jt a
9: satisfy := true;
10: for all c C do
11: if not ( )is_satisfy ,t c then
12: satisfy := false;
13: break;
14: if satisfy then
15: if ( )satisfy_query , at Q and ( )satisfy_query , dt Q then
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T, resolving
the corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
, as well as the list of subclass cardinalities N, attribute query
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
, and dependency query
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
.
Using the list of subclass cardinalities N, the algorithm resolves the CSP only for those subclasses, whose
cardinality is matched with one of the list N. The set of constraints C is used by the procedure
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
to verify the satisfiability of the constraint
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
for the subclass
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
, if the constraint is applicable to the
subclass. In other words, the procedure
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
resolves the CSP for particular subclass of the class T
and if the CSP is satisfiable, then the subclass is semantically consistent. It allows the algorithm constructs
only semantically consistent subclasses of the class T. For each such subclass of the class T, the algorithm
performs the additional filtration according to attribute query
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
and dependency query
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
, using for this
purpose the procedure
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
. As the result, the algorithm constructs all semantically consistent
subclasses of the homogeneous class of objects T, which have a certain cardinality and satisfy the attribute
and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks, firstly,
it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the
corresponding restrictions.
150
Моделі і засоби систем баз даних та знань
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In
the first case, we can construct all possible semantically consistent subclasses of the class T and then store them
in a database, using for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into
the database as a corresponding table, where a particular subclass property will be mapped in the corresponding
column of the table. Following that, we can use SQL to perform the information retrieval. However, such mapping
is applicable only to properties of the class that restricts the usability of the approach because methods can be parts
of structural and functional molecules of the class. In the second case, we can perform the information retrieval on
the fly, via dynamic filtering of constructed semantically consistent subclasses. To perform the filtering, we can use
any query language, which is applicable for the querying over homogeneous classes of objects. However, in this
case, we need either develop our own processor or adapt one of the appropriate ones to convert the selected query
language to object-oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute
query
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
and dependency query
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
, which describe the inclusion of desired attributes and dependencies, as
well as the exclusion of undesired ones. Let us consider a few examples of dynamic knowledge retrieval using
the homogeneous class of objects Tr defined above. Suppose we want to retrieve all semantically consistent
subclasses of the class Tr, which have a cardinality from 4 to 6, and contain attributes
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
, for this
purpose we need to set for Algorithm 1 the following configuration:
Моделі і засоби систем баз даних і знань
16: ( ).add ;D t
17: : {};t =
18: return D .
As we can see, Algorithm 1 performs the decomposition of the homogeneous class of objects T , resolving the
corresponding constraint satisfaction problem (CSP), using the set of its internal semantic dependencies ( )C ISD T= ,
as well as the list of subclass cardinalities N , attribute query aQ , and dependency query dQ . Using the list of subclass
cardinalities N , the algorithm resolves the CSP only for those subclasses, whose cardinality is matched with one of the
list N . The set of constraints C is used by the procedure ( )is_satisfy ,t c to verify the satisfiability of the constraint
c С for the subclass t T , if the constraint is applicable to the subclass. In other words, the procedure
( )is_satisfy ,t c resolves the CSP for particular subclass of the class T and if the CSP is satisfiable, then the subclass is
semantically consistent. It allows the algorithm constructs only semantically consistent subclasses of the class T . For
each such subclass of the class T , the algorithm performs the additional filtration according to attribute query aQ and
dependency query dQ , using for this purpose the procedure ( )satisfy_query ,t Q . As the result, the algorithm constructs
all semantically consistent subclasses of the homogeneous class of objects T , which have a certain cardinality and
satisfy the attribute and dependency restrictions, if such subclasses exist. In general, Algorithm 1 performs two tasks,
firstly, it extracts the conceptual knowledge via decomposition of homogeneous classes of objects onto the set of
semantically consistent subclasses, and secondly, it retrieves the particular subclasses, which satisfy the corresponding
restrictions.
Procedure 1. ( )is_satisfy ,t c
Input: ,t c
Output: satisfy {true, false, none}
1: satisfy := none;
2: if [0]c t then
3: for , 1,...,| |ic c i c = do
4: for [ ][ ] [ ], 1,...,| [ ] |c i j c i j c i = do
5: if [ ][ ]c i j t then
6: satisfy := true;
7: else
8: satisfy := false;
9: break;
10: if satisfy then
11: return satisfy;
12: return satisfy.
Procedure 2. ( )satisfy_query ,t Q
Input: ,t Q
Output: satisfy {true, false}
1: if | [0] | 0Q = and | [1] | 0Q = then
2: return true;
3: for [0]q Q do
4: if q t then
5: return false;
6: for [1]q Q do
7: if q t then
8: return false;
9: return true.
Consequently, there are two different scenarios for the organization of conceptual knowledge retrieval. In the first case,
we can construct all possible semantically consistent subclasses of the class T and then store them in a database, using
for this object-relational mapping. Indeed, according to [1-2], each class will be mapped into the database as a
corresponding table, where a particular subclass property will be mapped in the corresponding column of the table.
Following that, we can use SQL to perform the information retrieval. However, such mapping is applicable only to
properties of the class that restricts the usability of the approach because methods can be parts of structural and
functional molecules of the class. In the second case, we can perform the information retrieval on the fly, via dynamic
filtering of constructed semantically consistent subclasses. To perform the filtering, we can use any query language,
which is applicable for the querying over homogeneous classes of objects. However, in this case, we need either
develop our own processor or adapt one of the appropriate ones to convert the selected query language to object-
oriented structures.
To filter semantically consistent subclasses during the retrieval stage, we propose to use attribute query aQ and
dependency query dQ , which describe the inclusion of desired attributes and dependencies, as well as the exclusion of
undesired ones. Let us consider a few examples of dynamic knowledge retrieval using the homogeneous class of objects
Tr defined above. Suppose we want to retrieve all semantically consistent subclasses of the class Tr , which have a
cardinality from 4 to 6, and contain attributes 1.Tr p and 1.Tr f , for this purpose we need to set for Algorithm 1 the
following configuration:
( ) (
)
1 1 1, , 4,5,6 , . , . , ,
, .
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
,
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
. Suppose we want to retrieve all
semantically consistent subclasses of the class Tr, which have the same cardinality as previously, and do not contain
attributes
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
, for this purpose we need to set for Algorithm 1 the following configuration:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
As the result, we will receive the following list of subclasses:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
,
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
,
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
. If we join configurations
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
we will receive the following result:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
. Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr, which have a cardinality from 4 to 6, and contain
functional molecules
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
, for this purpose we need to set for Algorithm 1 the following configuration:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
In this case, the algorithm will return the following subclasses:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
,
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
,
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
. If we need to retrieve all semantically
consistent subclasses of the class Tr, which have a cardinality from 4 to 6, and do not contain functional molecules
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
, for this purpose we need to set for Algorithm 1 the following configuration:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
151
Моделі і засоби систем баз даних та знань
In this case, the algorithm will return the following subclasses:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
,
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
. If we join configurations
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
then we will receive the following result:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
. Finally, if we join
configurations
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
, i.e.
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
then the algorithm will return the following subclasses:
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
.
Let us consider the interpretation of the obtained results for configuration D7 in more detail. As we
can see, each of the subclasses
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
define two points on a plane with the ability to get
and set their coordinates, as well as compute the distance between them. The subclass
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
defines three
points on a plane with the ability to get and set their coordinates, as well as compute the distance between
any two of them. The subclass
Моделі і засоби систем баз даних і знань
[Введите текст]
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
3SC Tr , ( )4
4SC Tr , ( )4
7SC Tr , ( )4
8SC Tr ,
( )5
1SC Tr , ( )5
2SC Tr , ( )5
4SC Tr , ( )5
5SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . Suppose we want to retrieve all
semantically consistent subclasses of the class Tr , which have the same cardinality as previously, and do not contain
attributes 2.Tr f and 4.Tr f , for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
)
2 2 4, , 4,5,6 , , . , . ,
, .
a
d
D Tr ISD Tr N Q include exclude Tr f Tr f
Q include exclude
= = = = =
= = =
As the result, we will receive the following list of subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 1D and 2D , i.e.
( ) (
)
3 1 1 2 4, , 4,5,6 , . , . , . , . ,
, ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include exclude
= = = = =
= = =
we will receive the following result: ( )4
1SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Let us assume that we
need to retrieve all semantically consistent subclasses of the class Tr , which have a cardinality from 4 to 6, and contain
functional molecules ( )1FM Tr , ( )3FM Tr , for this purpose we need to set for Algorithm 1 the following
configuration:
( ) (
( ) ( ) )
4
1 3
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , ( )5
4SC Tr ,
( )5
5SC Tr , ( )5
6SC Tr , ( )6
1SC Tr , ( )6
2SC Tr , and ( )6
3SC Tr . If we need to retrieve all semantically consistent subclasses
of the class Tr , which have a cardinality from 4 to 6, and do not contain functional molecules ( )2FM Tr , ( )4FM Tr ,
for this purpose we need to set for Algorithm 1 the following configuration:
( ) (
( ) ( ) )
5
2 4
, , 4,5,6 , , ,
, , .
a
d
D Tr ISD Tr N Q include exclude
Q include exclude FM Tr FM Tr
= = = = =
= = =
In this case, the algorithm will return the following subclasses: ( )4
1SC Tr , ( )4
6SC Tr , ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr ,
( )5
2SC Tr , and ( )6
1SC Tr . If we join configurations 4D and 5D , i.e.
( ) (
( ) ( ) ( ) ( ) )
6
1 3 2 4
, , 4,5,6 , , ,
, , , ,
a
d
D Tr ISD Tr N Q include exclude
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then we will receive the following result: ( )4
7SC Tr , ( )4
8SC Tr , ( )4
9SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr . Finally, if we join
configurations 3D and 6D , i.e.
( ) (
( ) ( ) ( ) ( ) )
7 1 1 2 4
1 3 2 4
, , 4,5,6 , . , . , . , . ,
, , , ,
a
d
D Tr ISD Tr N Q include Tr p Tr f exclude Tr f Tr f
Q include FM Tr FM Tr exclude FM Tr FM Tr
= = = = =
= = =
then the algorithm will return the following subclasses: ( )4
7SC Tr , ( )4
8SC Tr , ( )5
2SC Tr , and ( )6
1SC Tr .
Let us consider the interpretation of the obtained results for configuration 7D in more detail. As we can see, each
of the subclasses ( )4
7SC Tr and ( )4
8SC Tr define two points on a plane with the ability to get and set their coordinates,
as well as compute the distance between them. The subclass ( )5
2SC Tr defines three points on a plane with the ability to
get and set their coordinates, as well as compute the distance between any two of them. The subclass ( )6
1SC Tr defines
a triangle on a plane with the ability to get and set coordinates of its vertices, as well as compute the length of all its
sides. Therefore, we can conclude that decomposition of the homogeneous class of objects Tr using Algorithm 1
generates semantically consistent subclasses, which represent the implicit or hidden knowledge within the domain of
defines a triangle on a plane with the ability to get and set coordinates
of its vertices, as well as compute the length of all its sides. Therefore, we can conclude that decomposition
of the homogeneous class of objects Tr using Algorithm 1 generates semantically consistent subclasses,
which represent the implicit or hidden knowledge within the domain of the class Tr. In addition, Algorithm 1
performs the filtration of all constructed semantically consistent subclasses according to attribute query
Моделі і засоби систем баз даних і знань
the class Tr . In addition, Algorithm 1 performs the filtration of all constructed semantically consistent subclasses
according to attribute query aQ and dependency query dQ .
For some cases, Algorithm 1 can be improved by changing the filtration strategy, since a resolving of
decompositional SCP is also a kind of subclass filtration, depending on an attribute query, as well as a dependency
query, the order of verification of their satisfiability can be changed. The main criterion for such modification of
Algorithm 1 is the estimation of search space reducing chain performed by a particular sequence of subclass filtration.
Conclusions
In this paper, we considered in detail the internal semantic dependencies of homogeneous classes of objects
(structural and functional atoms and molecules) and how they affect the decomposition of the class. We defined the
decomposition of the class as splitting the class into such subclasses, which do not contradict any internal semantic
dependency. Since all possible subclasses of a homogeneous class of objects form a power set lattice, which is a
complete lattice, using methods of formal concept analysis we constructed the corresponding concept lattices for all
subclasses of the class, for all internal semantic dependencies of the class, and for all its semantically consistent
subclasses. As the result, we found that in all three cases, constructed concept lattices contain a certain number of
formal concepts with semantically inconsistent intents because the algorithms for the construction of concept lattices
compute the part of extents via the intersection of extents which can be extracted from the formal context. At the same
time, they do not consider the internal semantic dependencies of a class, which define corresponding restrictions to the
creation of its semantically consistent subclasses. That restricts the usage of formal concept analysis for knowledge
extraction and retrieval since it allows retrieval, inference, or usage of inconsistent concepts, which are unreal within a
modeled domain.
To propose an alternative approach to knowledge extraction and retrieval via decomposition of homogeneous
classes of objects, we improved the decomposition algorithm, which was proposed in [28], adding the additional
filtering parameters, which help to reduce the search space and improve the performance. As the result, in the first
stage, the algorithm extracts knowledge by constructing only semantically consistent subclasses of a homogeneous class
of objects, which have a certain cardinality, via solving the corresponding constraint satisfaction problem defined based
on the internal semantic dependencies of the class. In the second stage, the algorithm retrieves knowledge by filtration
of constructed semantically consistent subclasses according to the attribute and dependency queries, which allow
selecting only those subclasses, which include all desired attributes and dependencies and do not include undesired
ones. We introduced the decomposition consistency coefficient, which allows us to estimate how much the algorithm
can reduce the search space for knowledge extraction and retrieval, avoiding the consideration of all semantically
inconsistent subclasses of the class. To demonstrate the possible application of the algorithm, we considered seven
different scenarios of how the homogeneous class of objects, which define a triangle on a plane, can be decomposed for
knowledge extraction and retrieval. In all cases, the algorithm extracted and retrieved subclasses of the class, which are
semantically consistent within a modeled domain and satisfy all restrictions and filters. However, despite all advantages
of the developed algorithm, it requires future analysis, improvement, and optimization.
Acknowledgments
This research has been supported by the National Academy of Science of Ukraine (project 0121U111944
Development of Methods and Tools for Construction of Domain-Oriented Intelligent Software Systems Based on
Object-Oriented Dynamic Networks).
References
1. AMBLER, S. W. (2003) Agile Database Techniques: Effective Strategies for the Agile Software Developer. Indianapolis, IN, USA: John
Willey & Sons, Ltd.
2. AMBLER, S. W. (2004) The Object Primer: Agile Model-Driven Development with UML 2.0. 3rd Ed. New York, NY, USA: Cambridge
University Press.
3. APT, K. R. (2003) Principles of Constraint Programming. New York, NY, USA: Cambridge University Press. doi:
https://doi.org/10.1017/CBO9780511615320
4. BAIXERIES, J., KAYTOUE, M. & NAPOLI, A. (2014) Characterizing functional dependencies in formal concept analysis with pattern
structures. Annals of Mathematics and Artificial Intelligence. 72 (1-2). pp. 129-149. doi: https://doi.org/10.1007/s10472-014-9400-3
5. BAIXERIES, J., KAYTOUE, M. & NAPOLI, A. (2014) Characterization of Database Dependencies with FCA and Pattern Structures. In:
IGNATOV, D. I. et al. (Eds.). Analysis of Images, Social Networks and Texts. AIST 2014. Communication in Computer and Information
Science, vol. 436. Switzerland AZ: Springer, Cham. doi: https://doi.org/10.1007/978-3-319-12580-0_1
6. BAIXERIES, J. et al. (2022) Computing Dependencies Using FCA. In: MISSAOUI, R., KWUIDA, L. & ABDESSALEM, T. (Eds.).
Complex Data Analytics with Formal Concept Analysis. Switzerland AZ: Springer Cham. doi: https://doi.org/10.1007/978-3-030-93278-
7_6
7. CARPINETO, C. & ROMANO, G. (2004) Concept data analysis: Theory and application. New York, NY, USA: John Willey & Sons,
Ltd.
8. CARUCCIO, L., DEUFEMIA, V. & POLESE, G. (2016) Relaxed Functional Dependencies – A Survey of Approaches. IEEE
Transactions on Knowledge and Data Engineering. 28 (1). pp. 147-165. doi: https://doi.org/10.1109/TKDE.2015.2472010
9. CODOCEDO, V. et al. (2013) Using pattern structures to support information retrieval with Formal Concept Analysis. In: Proceedings of
the International Workshop “What can FCA do for Artificial Intelligence?” (FCA4AI at IJCAI 2013). Beijing, China, 5 August 2013. pp.
15-24.
10. CODOCEDO, V., LYKOURENTZOU, I. & NAPOLI, A. (2014) A semantic approach to concept lattice-based information retrieval.
Annals of Mathematics and Artificial Intelligence. 72 (1-2). pp. 169-195. doi: https://doi.org/10.1007/s10472-014-9403-0
and dependency query
Моделі і засоби систем баз даних і знань
the class Tr . In addition, Algorithm 1 performs the filtration of all constructed semantically consistent subclasses
according to attribute query aQ and dependency query dQ .
For some cases, Algorithm 1 can be improved by changing the filtration strategy, since a resolving of
decompositional SCP is also a kind of subclass filtration, depending on an attribute query, as well as a dependency
query, the order of verification of their satisfiability can be changed. The main criterion for such modification of
Algorithm 1 is the estimation of search space reducing chain performed by a particular sequence of subclass filtration.
Conclusions
In this paper, we considered in detail the internal semantic dependencies of homogeneous classes of objects
(structural and functional atoms and molecules) and how they affect the decomposition of the class. We defined the
decomposition of the class as splitting the class into such subclasses, which do not contradict any internal semantic
dependency. Since all possible subclasses of a homogeneous class of objects form a power set lattice, which is a
complete lattice, using methods of formal concept analysis we constructed the corresponding concept lattices for all
subclasses of the class, for all internal semantic dependencies of the class, and for all its semantically consistent
subclasses. As the result, we found that in all three cases, constructed concept lattices contain a certain number of
formal concepts with semantically inconsistent intents because the algorithms for the construction of concept lattices
compute the part of extents via the intersection of extents which can be extracted from the formal context. At the same
time, they do not consider the internal semantic dependencies of a class, which define corresponding restrictions to the
creation of its semantically consistent subclasses. That restricts the usage of formal concept analysis for knowledge
extraction and retrieval since it allows retrieval, inference, or usage of inconsistent concepts, which are unreal within a
modeled domain.
To propose an alternative approach to knowledge extraction and retrieval via decomposition of homogeneous
classes of objects, we improved the decomposition algorithm, which was proposed in [28], adding the additional
filtering parameters, which help to reduce the search space and improve the performance. As the result, in the first
stage, the algorithm extracts knowledge by constructing only semantically consistent subclasses of a homogeneous class
of objects, which have a certain cardinality, via solving the corresponding constraint satisfaction problem defined based
on the internal semantic dependencies of the class. In the second stage, the algorithm retrieves knowledge by filtration
of constructed semantically consistent subclasses according to the attribute and dependency queries, which allow
selecting only those subclasses, which include all desired attributes and dependencies and do not include undesired
ones. We introduced the decomposition consistency coefficient, which allows us to estimate how much the algorithm
can reduce the search space for knowledge extraction and retrieval, avoiding the consideration of all semantically
inconsistent subclasses of the class. To demonstrate the possible application of the algorithm, we considered seven
different scenarios of how the homogeneous class of objects, which define a triangle on a plane, can be decomposed for
knowledge extraction and retrieval. In all cases, the algorithm extracted and retrieved subclasses of the class, which are
semantically consistent within a modeled domain and satisfy all restrictions and filters. However, despite all advantages
of the developed algorithm, it requires future analysis, improvement, and optimization.
Acknowledgments
This research has been supported by the National Academy of Science of Ukraine (project 0121U111944
Development of Methods and Tools for Construction of Domain-Oriented Intelligent Software Systems Based on
Object-Oriented Dynamic Networks).
References
1. AMBLER, S. W. (2003) Agile Database Techniques: Effective Strategies for the Agile Software Developer. Indianapolis, IN, USA: John
Willey & Sons, Ltd.
2. AMBLER, S. W. (2004) The Object Primer: Agile Model-Driven Development with UML 2.0. 3rd Ed. New York, NY, USA: Cambridge
University Press.
3. APT, K. R. (2003) Principles of Constraint Programming. New York, NY, USA: Cambridge University Press. doi:
https://doi.org/10.1017/CBO9780511615320
4. BAIXERIES, J., KAYTOUE, M. & NAPOLI, A. (2014) Characterizing functional dependencies in formal concept analysis with pattern
structures. Annals of Mathematics and Artificial Intelligence. 72 (1-2). pp. 129-149. doi: https://doi.org/10.1007/s10472-014-9400-3
5. BAIXERIES, J., KAYTOUE, M. & NAPOLI, A. (2014) Characterization of Database Dependencies with FCA and Pattern Structures. In:
IGNATOV, D. I. et al. (Eds.). Analysis of Images, Social Networks and Texts. AIST 2014. Communication in Computer and Information
Science, vol. 436. Switzerland AZ: Springer, Cham. doi: https://doi.org/10.1007/978-3-319-12580-0_1
6. BAIXERIES, J. et al. (2022) Computing Dependencies Using FCA. In: MISSAOUI, R., KWUIDA, L. & ABDESSALEM, T. (Eds.).
Complex Data Analytics with Formal Concept Analysis. Switzerland AZ: Springer Cham. doi: https://doi.org/10.1007/978-3-030-93278-
7_6
7. CARPINETO, C. & ROMANO, G. (2004) Concept data analysis: Theory and application. New York, NY, USA: John Willey & Sons,
Ltd.
8. CARUCCIO, L., DEUFEMIA, V. & POLESE, G. (2016) Relaxed Functional Dependencies – A Survey of Approaches. IEEE
Transactions on Knowledge and Data Engineering. 28 (1). pp. 147-165. doi: https://doi.org/10.1109/TKDE.2015.2472010
9. CODOCEDO, V. et al. (2013) Using pattern structures to support information retrieval with Formal Concept Analysis. In: Proceedings of
the International Workshop “What can FCA do for Artificial Intelligence?” (FCA4AI at IJCAI 2013). Beijing, China, 5 August 2013. pp.
15-24.
10. CODOCEDO, V., LYKOURENTZOU, I. & NAPOLI, A. (2014) A semantic approach to concept lattice-based information retrieval.
Annals of Mathematics and Artificial Intelligence. 72 (1-2). pp. 169-195. doi: https://doi.org/10.1007/s10472-014-9403-0
.
For some cases, Algorithm 1 can be improved by changing the filtration strategy, since a resolving of
decompositional SCP is also a kind of subclass filtration, depending on an attribute query, as well as a dependency
query, the order of verification of their satisfiability can be changed. The main criterion for such modification of
Algorithm 1 is the estimation of search space reducing chain performed by a particular sequence of subclass filtration.
Conclusions
In this paper, we considered in detail the internal semantic dependencies of homogeneous classes of
objects (structural and functional atoms and molecules) and how they affect the decomposition of the class. We
defined the decomposition of the class as splitting the class into such subclasses, which do not contradict any
internal semantic dependency. Since all possible subclasses of a homogeneous class of objects form a power set
lattice, which is a complete lattice, using methods of formal concept analysis we constructed the corresponding
concept lattices for all subclasses of the class, for all internal semantic dependencies of the class, and for all its
semantically consistent subclasses. As the result, we found that in all three cases, constructed concept lattices
contain a certain number of formal concepts with semantically inconsistent intents because the algorithms for the
construction of concept lattices compute the part of extents via the intersection of extents which can be extracted
from the formal context. At the same time, they do not consider the internal semantic dependencies of a class,
which define corresponding restrictions to the creation of its semantically consistent subclasses. That restricts
the usage of formal concept analysis for knowledge extraction and retrieval since it allows retrieval, inference, or
usage of inconsistent concepts, which are unreal within a modeled domain.
To propose an alternative approach to knowledge extraction and retrieval via decomposition of
homogeneous classes of objects, we improved the decomposition algorithm, which was proposed in [28], adding
the additional filtering parameters, which help to reduce the search space and improve the performance. As the
result, in the first stage, the algorithm extracts knowledge by constructing only semantically consistent subclasses
of a homogeneous class of objects, which have a certain cardinality, via solving the corresponding constraint
satisfaction problem defined based on the internal semantic dependencies of the class. In the second stage, the
algorithm retrieves knowledge by filtration of constructed semantically consistent subclasses according to the
attribute and dependency queries, which allow selecting only those subclasses, which include all desired attributes
and dependencies and do not include undesired ones. We introduced the decomposition consistency coefficient,
which allows us to estimate how much the algorithm can reduce the search space for knowledge extraction and
retrieval, avoiding the consideration of all semantically inconsistent subclasses of the class. To demonstrate
the possible application of the algorithm, we considered seven different scenarios of how the homogeneous
class of objects, which define a triangle on a plane, can be decomposed for knowledge extraction and retrieval.
In all cases, the algorithm extracted and retrieved subclasses of the class, which are semantically consistent
within a modeled domain and satisfy all restrictions and filters. However, despite all advantages of the developed
algorithm, it requires future analysis, improvement, and optimization.
Acknowledgments
This research has been supported by the National Academy of Science of Ukraine (project 0121U111944
Development of Methods and Tools for Construction of Domain-Oriented Intelligent Software Systems Based on
Object-Oriented Dynamic Networks).
152
Моделі і засоби систем баз даних та знань
References
1. AMBLER, S. W. (2003) Agile Database Techniques: Effective Strategies for the Agile Software Developer. Indianapolis, IN, USA: John
Willey & Sons, Ltd.
2. AMBLER, S. W. (2004) The Object Primer: Agile Model-Driven Development with UML 2.0. 3rd Ed. New York, NY, USA: Cambridge
University Press.
3. APT, K. R. (2003) Principles of Constraint Programming. New York, NY, USA: Cambridge University Press. doi: https://doi.org/10.1017/
CBO9780511615320
4. BAIXERIES, J., KAYTOUE, M. & NAPOLI, A. (2014) Characterizing functional dependencies in formal concept analysis with pattern
structures. Annals of Mathematics and Artificial Intelligence. 72 (1-2). pp. 129-149. doi: https://doi.org/10.1007/s10472-014-9400-3
5. BAIXERIES, J., KAYTOUE, M. & NAPOLI, A. (2014) Characterization of Database Dependencies with FCA and Pattern Structures. In:
IGNATOV, D. I. et al. (Eds.). Analysis of Images, Social Networks and Texts. AIST 2014. Communication in Computer and Information
Science, vol. 436. Switzerland AZ: Springer, Cham. doi: https://doi.org/10.1007/978-3-319-12580-0_1
6. BAIXERIES, J. et al. (2022) Computing Dependencies Using FCA. In: MISSAOUI, R., KWUIDA, L. & ABDESSALEM, T. (Eds.).
Complex Data Analytics with Formal Concept Analysis. Switzerland AZ: Springer Cham. doi: https://doi.org/10.1007/978-3-030-93278-
7_6
7. CARPINETO, C. & ROMANO, G. (2004) Concept data analysis: Theory and application. New York, NY, USA: John Willey & Sons, Ltd.
8. CARUCCIO, L., DEUFEMIA, V. & POLESE, G. (2016) Relaxed Functional Dependencies – A Survey of Approaches. IEEE Transactions
on Knowledge and Data Engineering. 28 (1). pp. 147-165. doi: https://doi.org/10.1109/TKDE.2015.2472010
9. CODOCEDO, V. et al. (2013) Using pattern structures to support information retrieval with Formal Concept Analysis. In: Proceedings of
the International Workshop “What can FCA do for Artificial Intelligence?” (FCA4AI at IJCAI 2013). Beijing, China, 5 August 2013. pp.
15-24.
10. CODOCEDO, V., LYKOURENTZOU, I. & NAPOLI, A. (2014) A semantic approach to concept lattice-based information retrieval. Annals
of Mathematics and Artificial Intelligence. 72 (1-2). pp. 169-195. doi: https://doi.org/10.1007/s10472-014-9403-0
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Received 11.08.2022
153
Моделі і засоби систем баз даних та знань
About authors:
Dmytro O. Terletskyi
Ph.D. in Computer Science
Senior Research Fellow
ORCID: 0000-0002-7393-1426
Sergiy V. Yershov
D.Sc. in Computer Science
Senior Research Fellow
Scientific Secretary
ORCID: 0000-0002-9895-777X
Place of work:
V. M. Glushkov Institute of Cybernetics of National Academy of Science of Ukraine
Kyiv-03187, 40 Academician Glushkov Avenue
Tel.: +380 (44) 526-64-22
Email: dmytro.terletskyi@nas.gov.ua
ErshovSV@nas.gov.ua
Прізвища та ініціали авторів і назва доповіді українською мовою:
Д. О. Терлецький, С. В. Єршов
Декомпозиційне видобування та пошук концептуальних знань
Прізвища та ініціали авторів і назва доповіді англійською мовою:
Dmytro O. Terletskyi, Sergiy V. Yershov
Decompositional Extraction and Retrieval of Conceptual Knowledge
|