The Non-Deterministic Strategy of Rewriting
Описана система алгебраического программирования – первая система переписывания термов, разделившая системы переписывающих правил и стратегии, а также система моделирования – базовая для системы верификации формальных спецификаций. The Algebraic Programming System is described – the first term rewri...
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
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| Cite this: | The Non-Deterministic Strategy of Rewriting / A.A. Letichevsky, O.A. Letichevskyi, V.S. Peschanenko // Управляющие системы и машины. — 2013. — № 6. — С. 53-58. — Бібліогр.: 15 назв. — англ. |
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| author | Letichevsky, A.A. Letichevskyi, O.A. Peschanenko, V.S. |
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| citation_txt | The Non-Deterministic Strategy of Rewriting / A.A. Letichevsky, O.A. Letichevskyi, V.S. Peschanenko // Управляющие системы и машины. — 2013. — № 6. — С. 53-58. — Бібліогр.: 15 назв. — англ. |
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| description | Описана система алгебраического программирования – первая система переписывания термов, разделившая системы переписывающих правил и стратегии, а также система моделирования – базовая для системы верификации формальных спецификаций.
The Algebraic Programming System is described – the first term rewriting system, which uses the rewriting rules system and strategies separately and to Insertion Modeling System IMS – a basic system for the Verification of a Formal Specification system.
Описано систему алгебраїчного програмування – першу систему переписування термів, яка розділила системи правил перепи- сування і стратегії, та систему інсерційного моделювання – базову для системи верифікації формальних специфікацій
|
| first_indexed | 2025-11-28T13:04:20Z |
| format | Article |
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УСиМ, 2013, № 6 53
Информационные технологии и системы
UDC 004.41,004.51
A.A. Letichevsky, O.A. Letichevskyi, V.S. Peschanenko
The Non-Deterministic Strategy of Rewriting
Описана система алгебраического программирования – первая система переписывания термов, разделившая системы переписываю-
щих правил и стратегии, а также система моделирования – базовая для системы верификации формальных спецификаций.
The Algebraic Programming System is described – the first term rewriting system, which uses the rewriting rules system and strategies
separately and to Insertion Modeling System IMS – a basic system for the Verification of a Formal Specification system.
Описано систему алгебраїчного програмування – першу систему переписування термів, яка розділила системи правил перепи-
сування і стратегії, та систему інсерційного моделювання – базову для системи верифікації формальних специфікацій.
Introduction. Algebraic Programming System APS
[1] was developed by the departments 100, 105 of
Glushkov Institute of Cybernetics of the National
Academy of Science of Ukraine [2] in 1987. It
was the first system of term rewriting which used
the RRS and strategies separately. The last version
of APS system was created with collaboration of
Research Institute of Information Technologies of
Kherson State University [3] in 2009.
Unlike traditional approach oriented to the us-
age of canonical RRS with “transparent” strategy
of their application, in APS it is possible to com-
bine arbitrary RRS with different strategies of re-
writing. Such approach essentially extends the
possibilities of rewriting technique enlarging the
flexibility and expressiveness of it. The APS inte-
grates four main programming paradigms in the
following way. The main part of the program can
be written in the form of rewriting systems. Im-
perative and functional programming is used for
the definition of strategies. Logic paradigm is re-
alized on a base of rewriting using built-in unifi-
cation procedure. One of its most important appli-
cation is the Insertion Modeling System (IMS).
The main differences between APS and ELAN [4],
MAUDE [5], STRATEGO [6] are presented.
So, the article is devoted to a special rewriting
strategy of APS system and its application for
VFS system.
The section “APS and Other” describes the
functional possibilities and differences between
APS system and ELAN, MAUDE, STRATEGO
systems. The section “APS Rewriting” is devoted
to RRS representation in APS and some special
algorithms of work with RRS. We describe our
proposal for non-deterministic rewriting strategy
(ND Strategy) in section “Non-Deterministic Stra-
tegy of Rewriting”. In section “VFS system” we
propose a short description of Semantics of Basic
Protocols of VFS system. The section “Non-
Deterministic Strategy Application” presents ex-
ample of application of such special strategy for
verification.
APS and Other
Let's demonstrate the comparison of functional
possibilities of APS system [1] and ELAN [2],
MAUDE [3], STRATEGO [4] systems.
Where EA, SA is transformation systems in-
cluding the compilers, interpreters, static analyz-
ers, domain-specific optimizers, code generators,
source code refectories, documentation genera-
tors, and document transformers; MA is the gen-
eral logics and logical frameworks, specification
languages, declarative programming languages,
semantics of programming languages and models
of computation, distributed systems, formal tools
and formal interoperability, reflection and meta-
programming, object-oriented modelling and pro-
gramming, real-time systems, bio informatics,
mobile languages, network protocols and active
networks; AA – algebraic programming, inser-
tional modelling, program transformation, general
logics and logical frameworks, specification lan-
guages, declarative programming; ACP – VRS
54 УСиМ, 2013, № 6
(Verification of Requirement Specification), TERM
(School System of Computer Algebra); * – can't
find any information about concrete projects; ** –
to the binary files and system commands; *** – C –
version of the arbitrary paths of program; +,–
means that the system supports compilation of
some small sub-set of system's language.
Without doubts due to quite developed typi-
fication in the system, MAUDE has more benefits
than other systems (the process of evaluation with
integers numbers). In this connection, there is a
quite limited number of rewriting strategies in the
system that considerably complicates the algo-
rithms realized in it.
It is clear from the table that APS doesn't con-
cede to well-known systems of terms rewriting as
per all criteria.
Let's compare the capacity of terms rewriting
systems taking the example of finding of n –
number of Fibonacci (in this case we are inter-
ested in total operation time of the program which
is used for rewriting only).
We are going to perform test on DELL
VOSTRO 1500 (CPU Intel Core duo 2.0, Memory
2 Gb, HDD 160 Gb). The results of launching of
this program in different systems of term rewriting
are presented:
From the other side, APS was considered to
be one of the slowest systems of term rewriting.
Taking into consideration the results of capacity
of rewriting, it can be said that APS is a quite
quick system of term rewriting (after the elimi-
nation of some deficiencies). Surely, it doesn't
have compiler, but instead of it we propose a
number of tools for convenience of program-
ming in APLAN (Algebraic Programming LAN-
guage, the language of APS system) as well as
in C++. These tools take APS to the new more
qualified level (the compilers of ELAN, MAUDE,
Stratego don't support language possibilities
completely).
T a b l e 2. The results of launching of algorithms finding of Fibo-
nacci n-number
Fibonacci number (in seconds)
No System names
15 20 21 22 23 24
1 Interpreter of ELAN 0 2 6 11,5 18,5 28
2
Interpreter of
Stratego
0 3 7 12 21 34
3
Interpreter of
MAUDE
0,004 0,04 0,068 0,072 0,104 0,236
4 Procedures of APS 0 1 1 3 4 7
5
Rewriting systems of
APS
0 2 2 4 6 10
Let's examine in details the mentioned above
deficiencies of APS system. The first version of
APS system had the memory leaks. The “garb”
operator for collection of waste in the program
was realized in APS system but as practice proved
this operator didn't delete memory completely. In
a course of analysis of the source code the design-
ers discovered places of memory leaks. It was
precipitated out 7 bytes of memory at the proce-
dure calling, described in APLAN language.
However, the actual C++ code which performs the
calling of these procedures does not contain obvi-
ous calling functions of memory selection. It led
to the fact that the used memory increased very
fast at execution of the program and some small
instances simply were terminated due to lack of
memory in computer. As a result we have taken the
following decision to implement the technology
Smart Pointers in APS [12]. Thus, by means of
this technology in the second version of APS sys-
tem it was possible to be saved of this deficiency.
T a b l e 1. Comparison of Functional Possibilities Between TRS’s
No Name
Strate-
gies
Number
None
Typing
Strategies
and rules
Proce-
dural
Lan-
guage
Possibili-
ties of
Language
Extension
User
Manual
Publica-
tion
Connec-
tion to the
External
Modules
Compila-
tion
Dynamical
Creation of
the RRS
Sup-
port
Applica-
tion Area
Commer-
cial Prod-
ucts
Country
1 ELAN arbitrary – – + 1992 – –,+ – + EA * France
2
STRA-
TEGO
arbitrary + – – 1994 – –,+ + – SA *
Nether-
lands
3 MAUDE 7 + – – 1995 – –,+ – + MA * USA
4 APS arbitrary + + + 1987 ** *** + + AA ACP Ukraine
УСиМ, 2013, № 6 55
APS Rewriting
General definition of syntax of RRS is the fol-
lowing:
< rewriting system >::= rs(< list o f variables
separated by ”,” >)
(< list o f rules separated by ”,” >)
< rule >::=< simple rule > | < conditional rule >
< simple rule >::=< algebraic expression >=<
algebraic expression >
< conditional rule >::=< condition > −>< sim-
ple rule >
< variable >::=< identifier >
Each application of RRS in APS satisfies the
following conditions now:
1. One of the rules of the system is applied or
arithmetic operation is performed at each step of
rewriting.
2. The choice of a rule is made according to the
sequence in which the rules have been written.
Each RRS in APS applies to a term )(ZTt ,
where )(ZT is algebra of terms of some alge-
braic program (see subsection, is a signature of
operation of this algebra and Z is a generic set of
terms with zero arities, with some strategy: applr
applies RRS to a term ones, appls applies while
it's possible etc.
Rewriting machine of APS realized strategy
applr which is a base for all strategies in APS.
APS used a special language for faster application
of REM – REM (REwriting Machine) lan-
guage[13]. Algebraic definition of REM language
is presented by the next algebra.
Let )(ZT be the base algebra of terms for
some algebraic program. Set is an operation
signature of this algebra, Z – is a generating set of
terms of arity 0. REM programs produce many-
sorted algebra ZkkRR above )( VZT , whe-
re V – is a set of variables of this program, kR –
set of programs of rank k. The corresponding sig-
nature contains the next operation:
1. ;0,: kRRR kkk
2. 1( ((),..., ())) : ,n m ntest R R n
0, 0, , ( )m m ART (function ART re-
turns arity of term).
3. VZTtnRRtmatch nn ,0,:)( 1 .
4. VZTtRtrewrite ,)( 0 .
5. VZTtuRtrewriteuIf ,;, 0 .
6. VZTkk StkRRthash
,0,:)( , where
VZTS
is set of marks and term of arity 0 from
T Z V .
Operations 4 and 5 have zero arity and they
produce elements for algebra of REM language, as
follows from definitions.
SSR which was successfully converted in repre-
sentation of REM language is called REM-program.
Dynamical Adding and Removing of Rules
from the RRS
The process of application of RRS to current
term has a few stages: conversion to REM-pro-
gram (one time for each system only), interpreting
of REM-program by the kernel of rewriting ma-
chine. It means that if we have to update RRS
(add, remove or update some rule), APS system
has to rebuild it into REM-program each time. But
this operation demands more time, especially for
big RRS. So, in APS system we have realized two
operators:
remove_rule_rs – the function for removing
of rule from RRS without rebuilding of it.
add_rule_rs – the function for adding of rule
to RRS without rebuilding, dynamical adding of
rule in REM language.
More interesting function is add_rule_rs.
This function adds new rule into RRS without
its rebuilding. Its means that we should add new
operation from signature of algebra which cor-
responds to insertion of rule into REM-program.
We should create new RRS by the next ideas to
this effect:
1. Using list of variables from current RRS for
new RRS with one new rule.
2. Conversion of new RRS into REM-program.
3. Insertion of new REM-program into current
RRS (there are two possibilities to add new rule into
RSS: to make it the first and to make it the last).
4. Using already known size of previous RSS
making of number of insertion rewriting rule (this
number can be used by appls strategy).
56 УСиМ, 2013, № 6
The unification algorithm of two REM-pro-
grams uses the next ideas (new_rs – new REM-
program, old_rs – old REM-program):
1. In any case new_rs will have the next tem-
plate match(new_rs_m)rewrite(_) or match(new_
rs_m) If(_,_).
2. If old_rs has the next template (in terms of
APS) hash(_)_ then if hash contains the main
mark of new_rs_m then we should call recursive
to hash(type(new_rs_m)) and new_rs if it's possi-
ble or if not then we should use + operator from
signature of algebra.
3. If old_rs has the next template _+_ then we
should try to add in first argument and if it is not
possible then to try to add in second argument.
4. If old_rs has the next template match(old_
rs_m)last_part (it is possible only in one case: if
body of match of old_rs and new_rs has equal
main mark) then we should eliminate the
new_rs_m from old_rs_m and call algorithm re-
cursively with last_part.
5. At last, we should use hash operator if it's
possible or + operator if it's not.
Non-Deterministic Strategy
A non-deterministic rewriting strategy is an
enhancement of theory of Set Functions for Func-
tional Logic Programming [14] by the means of
fuzzy sets and its application of rewriting. We use
the next conditions for realization of non-deter-
ministic rewriting strategy:
1. After each successful application of some
rule from RRS rewriting we continue with rules
written below current.
2. Results of application of non-deterministic
strategy of rewriting are separated by the special
non-deterministic operation +.
Let’s show how it differs from the applr strat-
egy. The strategy applr is presented as a function
with two arguments: term which should be rewrit-
ten and REM-program. The high level of realiza-
tion applr strategy in APLAN language is napplr:
, ;
: ( , ) ( , , )(
( , );
: ( , , ) ;
NAME napplr appl
napplr proc t p loc pr Yes s
let p rs pr p
s p t Nil pr Nil
( , );
( ,1: );
:
);
appls s appl
let s s
yes t s
The main part of it is presented by RRS appl,
which is applied interactively to a state of re-
writing machine s. The initial state contains
program p, initial term t joins with constant Nil,
and array pt which consists of «_». It is a pre-
condition which defines requirement to applr.
The postcondition of it is the following: if some
system R which corresponds to a program p is
applicable to a term t, then after stopping
s = (1 : R(t)). If not then s = 0 after stopping.
The high level of realization of applr strategy in
APLAN language is appl:
);
),,(),,(
,),,())(,,(),,(
),:1():1(
)(,,,,(:
rprtpperformrprtp
rprtqprnewtprprtqp
trt
prtrqprsappl
The information about functions appls, applr,
napplr, appl, perform, new is represented in [15].
For execution of these conditions we should
rebuild the result s after application of appl and
to add non-deterministic application of RRS
appl to r. So, let’s consider non-deterministic
rewriting strategy nds_napplr in APLAN lan-
guage:
_ , , _ , _ ;NAMES nds applr elm_colon nds appl nds appls
_ : ( , ) ( , , )(
( , );
: ( , , ) ;
( , _ );
_ ( );
:
);
nds napplr proc t p loc pr Yes s
let p rs pr p
s p t Nil pr Nil
appls s nds appl
elm colon s
yes t s
);
:
),(_)(_
)(,(:_
yyx
ycolonelmxcolonelmyx
yxrscolonelm
УСиМ, 2013, № 6 57
);
),,(),,(
,),,())(,,(),,(
),r,:1(_):1(
)(,,,,(:_
rprtpperformкprtp
rprtqprnewtprprtqp
tapplsndsrt
prtrqprsapplnds
);
)_(
);nds_appl,(
)(,_(:_
otherresndsreturn
otherappls
otherresndsprocappsnds
The optimization of ND Strategy is a very im-
portant thing, because if RRS is bigger then
checking all of non-deterministic choices will be
considerably slower. So, the main question is
whether such ND strategy is optimal or not?
Theorem of optimization of ND Strategy. All
non-deterministic simultaneously impossible cases
will be eliminated from a consideration of REM-
programs on a one step of interpretation with the
help of hash operator of signature Ω.
Let's consider simultaneously impossible cases
on a one step interpretation: different mark of a
term t of set of terms with arity 0. But all of those
cases will be in hash operator (it follows from its
definition). It means that we choose only one case
from all simultaneously impossible cases for cur-
rent step of interpretation. So, the current realiza-
tion of ND Strategy is optimal.
Semantics of Basic Protocols of VFS system
Each basic protocol is a Hoare triple P ,
where P is a process, and are precondition and
postcondition of process P, respectively. and
are represented by logical expressions of the base
language and define conditions on the set of states
of a system. A process of basic protocol is a finite
convergent process over the set C of environment
actions, which may contain the set A of agent ac-
tions. We shall use the following notation for arbi-
trary basic protocols: pre(b) = , post(b) = , and
the process of B is denoted as Pb .
Each basic protocol defines properties of the sys-
tem and can be understood as a statement of tempo-
ral logic: if the precondition is true then the process
of a protocol can start, and after it is successfully
terminated, the postcondition must be true [11].
Non-Deterministic Rewriting Strategy Ap-
plication for Verification
Let | 1,...,n i i iB P i n be a set
of basic protocols of a project, a predicate trans-
former ,Tr is a function defined on formulae
of the base language returning a new formula such
that ,Tr e , e – define formulae. A predi-
cate transformer strengthens the postcondition of a
basic protocol by adding residual properties from
the precondition.
We can represent a process of one-step applica-
tion of each protocol from set Bn to formulae e by
the ND Strategy and the following RRS:
1 1 1 1
n
( , )(
( ) e P ( , ),
,
( ) e P ( , )
);
a
n n n
S rs e u
sat e Tr e
sat e Tr e
where n – a number of basic protocols in project,
function sat checks satisfiability of conjunction of
precondition and current environment.
Other good example for application of ND Strat-
egy in verification is based on experiments for effec-
tive term hashing algorithm realization on APS.
Let niee in ,...,1| be a set of formulae,
| 1,...,n i i iB P i n be a set of basic
protocols of a project. How we could determine
the set of basic protocols Bi Bn from which we
could get state ei. To determine the set of basic
protocols Bi we can build the next RRS which
should be dynamically update after each step of
application of basic protocol:
Ba := rs(x)(
00 ( ) ,
,
( )
);
n
i
n i
e x B
e x B
Conclusion
The system of algebraic programming APS ex-
ceeds the majority of criteria of well-known TRS.
Among these criteria we can outline the most two
important ones: the presence of procedural lan-
58 УСиМ, 2013, № 6
guage (allows using simultaneously the paradigms
of declarative and imperative languages) and the
commercial usage (usage of system in real big
commercial products and not only in different re-
searches).
The non-deterministic rewriting strategy is op-
timal and together with REM-program updating
functions can be applied in different arias of APS
and IMS systems applications.
1. Algebraic Programing System Site. – access mode:
http://apsystem.org.ua
2. Department of Theory of Digital Automatic Machines
of Glushkov Institute of Cybernetics of National Acad-
emy of Science of Ukraine. – http:// icyb.kiev.ua
3. Laboratory for Development and Implementation Peda-
gogical Software of Research Institute of Information
Technologies of Kherson State Univeristy. – http://riit.
ksu.ks.ua/ index.php?q=en/node/88
4. Elan System Official Site. – http://elan. loria.fr
5. Maude System Official Site. – http:// maude.cs.uiuc.edu
6. Stratego System Official Site. – http:// www.program-
transformation.org/ Stratego/WebHome
7. Insertion programming / V.A. Volkov, A.A. Letichev-
sky, Y.V. Kapitonova et al. // Cybernetics and System
Analysis. – 2003. – N 3. – P. 19–32.
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tichevsky, Y. Kapitonova, V. Kotlyarov et al. // Problems
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Y. Kapitonova, V. Kotlyarov et al. // Cybernetics and
System Analysis. – 2002. – N 4. – P. 3–14.
10. Evidence algorithm and problems of representation
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/ A. Letichevsky, A. Degtyarev, J. Kapitonova et al. //
Proc. of the International Workshop on Logic and
Complexity in Comp. Sci. – Workshop. – University
Paris 12, Creteil, France. – 2001. – P. 159–167.
11. Systems specification by basic protocols / A. Leti-
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and System Analysis. – 2005. – N 4. – P. 3–21.
12. Yonat Sharon Smart pointers – what, why, which? –
http://ootips.org/ yonat/4dev/smart-pointers. html
13. Letichevsky A., Khomenko V. A rewriting machine and
optimization of strategies or term rewriting // Cyber-
netics and System Analysis. – 2002. – N 5. – P. 3–17.
14. Hanus M., Antoy S. Set functions for functional logic
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putations in aps // Theor, Comp. Sci. – 1993. – N 119. –
P. 145–171.
Поступила 26.06.2012
Тел. для справок: +38 095 324-1557 (Киев, Херсон)
E-mail: vladim@ksu.ks.ua, vladimirius@gmail.com
© A.A. Letichevsky, O.A. Letichevskyi Jr.,
V.S. Peschanenko, 2013
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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| id | nasplib_isofts_kiev_ua-123456789-83220 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0130-5395 |
| language | English |
| last_indexed | 2025-11-28T13:04:20Z |
| publishDate | 2013 |
| publisher | Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України |
| record_format | dspace |
| spelling | Letichevsky, A.A. Letichevskyi, O.A. Peschanenko, V.S. 2015-06-16T20:18:19Z 2015-06-16T20:18:19Z 2013 The Non-Deterministic Strategy of Rewriting / A.A. Letichevsky, O.A. Letichevskyi, V.S. Peschanenko // Управляющие системы и машины. — 2013. — № 6. — С. 53-58. — Бібліогр.: 15 назв. — англ. 0130-5395 https://nasplib.isofts.kiev.ua/handle/123456789/83220 004.41,004.51 Описана система алгебраического программирования – первая система переписывания термов, разделившая системы переписывающих правил и стратегии, а также система моделирования – базовая для системы верификации формальных спецификаций. The Algebraic Programming System is described – the first term rewriting system, which uses the rewriting rules system and strategies separately and to Insertion Modeling System IMS – a basic system for the Verification of a Formal Specification system. Описано систему алгебраїчного програмування – першу систему переписування термів, яка розділила системи правил перепи- сування і стратегії, та систему інсерційного моделювання – базову для системи верифікації формальних специфікацій en Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України Управляющие системы и машины Информационные технологии и системы The Non-Deterministic Strategy of Rewriting Недетермінована стратегія переписування Недетерминированная стратегия переписывания Article published earlier |
| spellingShingle | The Non-Deterministic Strategy of Rewriting Letichevsky, A.A. Letichevskyi, O.A. Peschanenko, V.S. Информационные технологии и системы |
| title | The Non-Deterministic Strategy of Rewriting |
| title_alt | Недетермінована стратегія переписування Недетерминированная стратегия переписывания |
| title_full | The Non-Deterministic Strategy of Rewriting |
| title_fullStr | The Non-Deterministic Strategy of Rewriting |
| title_full_unstemmed | The Non-Deterministic Strategy of Rewriting |
| title_short | The Non-Deterministic Strategy of Rewriting |
| title_sort | non-deterministic strategy of rewriting |
| topic | Информационные технологии и системы |
| topic_facet | Информационные технологии и системы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/83220 |
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