H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... the rewriting process [18, 19] This group and their collaborators have developed a very impressive collection of examples and case studies in areas such as logic programming languages, constraint solving, higher order substitution, equational theorem proving and other such computational systems [79, 160, 81, 16, 26, 145, 61, 62]. Besides the ELAN interpreter, there is ongoing work on compilation techniques, including compilation of ACrewriting [130] A nice feature of rewriting logic, namely its natural way of dealing with concurrency and interaction, is exploited by Viry to treat input output for ELAN within the logic ....

....by rewriting strategies. The researchers in the ELAN group have developed an impressive collection of rewriting logic specifications for different automated deduction inference systems, including logical languages, unification and narrowing [79, 160] Knuth Bendix completion with constraints [81], higher order unification [16] combination of unification algorithms [145] constraint solving [26] and termination and tree automata techniques [61, 62] In a somewhat similar vein, the work of Levy and Agust i [96, 95, 97] and of Schorlemmer [148] explores the relationships between rewriting ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, 1995.

.... to the sustained e orts of many researchers, particularly in the ELAN, Pisa, Stanford, and Maude teams, there is by now very extensive evidence supporting the claim that rewriting logic is indeed a 5 Clavel et al. very exible and simple semantic framework [21,23,24,4] and logical framework [20,18,34,19,3,32,6,14,7,10]. Moreover, object oriented design languages, architectural description languages, and languages for distributed components also have a natural semantics in rewriting logic [35,25,33,27,28] What is common to all these applications is that the models of computation, logics, or languages are ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, LNCS 914, pages 438-443. Springer-Verlag, 1995.

.... of rewriting logic are like two sides of the same coin, the same reasons making it a good semantic framework at the computational level make it also a good logical framework at the logical level, that is, a metalogic in which many other logics can be naturally represented and implemented [40, 39, 36, 37, 3, 57, 10, 22, 11, 17]. Moreover, object oriented design languages, architectural description languages, and languages for distributed components also have a natural semantics in rewriting logic [61, 51, 58, 53, 54] Consequently, some of the most interesting applications of Maude are metalanguage applications, in ....

Helene Kirchner and Pierre-Etienne Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, 1995.

.... have developed an impressive collection of rewriting logic specifications for different automated deduction inference systems, including the already mentioned theorem proving modulo methods [109 111] logical languages, unification and narrowing [169,320] Knuth Bendix completion with constraints [176], higherorder unification [15] combination of unification algorithms [277] constraint solving [46 50] and termination and tree automata techniques [149,150] In a somewhat similar vein, the work of Schorlemmer explores the relationships between rewriting logic and Levy and Agust i s general ....

H'el`ene Kirchner and Pierre-Etienne Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Rewriting Techniques and Applications, 6th International Conference, RTA'95, Kaiserslautern, Germany, April 5--7,

.... to the sustained efforts of many researchers, particularly in the ELAN, Pisa, Stanford, and Maude teams, there is by now very extensive evidence supporting the claim that rewriting logic is indeed a 5 Clavel et al. very flexible and simple semantic framework [21,23,24,4] and logical framework [20,18,34,19,3,32,6,14,7,10]. Moreover, object oriented design languages, architectural description languages, and languages for distributed components also have a natural semantics in rewriting logic [35,25,33,27,28] What is common to all these applications is that the models of computation, logics, or languages are ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, LNCS 914, pages 438--443. Springer-Verlag, 1995.

....are instead first order. Our work should of course be placed within the context of the above related work, and of experiments carried out in different frameworks to prototype formal systems (for more discussion see the survey [44] 2. 1 Rewriting Logic and Reflection We and other researchers [37, 36, 44, 33, 52, 34, 5, 49, 8, 10, 12] have investigated the suitability of rewriting logic [40] as a logical framework and have found it to have very good properties for this purpose. One important practical advantage is that, what might be called the representational distance between a theory T in the original logic and its ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, 1995.

.... confluence, combination and enrichment of computational systems [54] The framework has been used for prototyping and experimentating with several logics on interest, among which: i) our work on higher order unification via explicit substitutions [29] ii) completion with constraints [56]. We are now working on a more expressive strategy language and on the design and implementation of a compiler for the language. 7.2 Cooperations, visits, lectures, seminars within the working group Hubert Comon visited Nancy several times, giving a seminar on Sequentiality, second order ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany) , volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

.... Thanks to the sustained efforts of many researchers, particularly in the ELAN, Pisa, Stanford, and Maude teams, there is by now very extensive evidence supporting the claim that rewriting logic is indeed a very flexible and simple semantic framework [37, 40, 42, 6] and logical framework [32, 29, 62, 30, 2, 55, 8, 16, 10, 12]. Moreover, object oriented design languages, CHAPTER 1. INTRODUCTION 7 architectural description languages, and languages for distributed components also have a natural semantics in rewriting logic [63, 34, 57, 47, 48] see Section 2.8.2 for more discussion on the use of reflection in logical and ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, 1995.

....to guide the rewriting process. This group and their collaborators have developed a very impressive collection of examples and case studies in areas such as logic programming languages, constraint solving, higher order substitution, equational theorem proving and other such computational systems [46, 103, 49, 14, 18]. A nice feature of rewriting logic, namely its natural way of dealing with concurrency and interaction, is exploited by Viry to treat input output for ELAN within the logic itself [101] Futatsugi and Sawada at Japan Advanced Institute of Science and Technology (JAIST) in Kanazawa are in a very ....

.... justice to this logical side is beyond the scope of this paper, but good evidence, including a good number of examples of logic representations can be found in two joint papers with Mart i Oliet [68, 69] Additional quite impressive evidence is also provided by research based on the ELAN language [46, 103, 49, 14, 18], that stresses the logical framework applications of rewriting logic. There is also a very fruitful relationship between rewriting logic and the theory of reasoning theories proposed by Giunchiglia, Pecchiari and Talcott [39] Reasoning theories provide a logic independent architecture for ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proc. Rewriting Techniques and Applications, Kaiserslautern, 1995.

....issues pertaining to strategy and control. As we shall see, this is precisely what reflection and a convenient internal strategy language allow us to accomplish. There have been several attempts to use this abstract view of completion in actual rewrite rule implementations of completion procedures [22, 20]. The challenge has been to control the rewriting inference process generated by the rules in C which in principle could go in many undesired directions by means of adequate strategies. Unfortunately, these implementations have not been able to preserve the separation between the logical and ....

H. Kirchner and P. E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

....treats the case of unsorted and unconditional rewrite theories; the extension to sorted and conditional theories is unproblematic and will appear elsewhere. There have been several attempts to use this abstract view of completion in actual rewrite rule implementations of completion procedures [15, 13]. The challenge has been to control the rewriting inference process generated by the rules in C which in principle could go in many undesired directions by means of adequate strategies. Unfortunately, these implementations have not been able to preserve the separation between the logical and ....

H. Kirchner and P. E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

....and merging their ELAN sources. In a more general case, these applications are even not necessarily specified in ELAN. A procedure looking for general path orderings (gpo) for rewrite systems [9] is the first integrated application. The second one is the Knuth Bendix s completion procedure [15], which completes rewrite systems using the general path ordering found before. These applications do not need to be modified. Other applications, even not written in ELAN, may be combined in a similar way. For example, we could replace the ELAN implementation of the procedure gpo by its ECLIPSE ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany), volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

....was not aimed at e ciency but rather at studying di erent strategies. Thanks to the powerful strategy language of ELAN, it appears very easy to experiment various possibilities in a exible and high level way. 2 Completion with constraints Full de nitions and further references can be found in [2]. A constraint is a rstorder formula built on a signature . Elementary constraints are two constants T and F , and equations (t = t 0 ) with t; t 0 in the set of terms T ( X ) Nonelementary constraints are obtained by conjunction. The notation c is used to denote the substitution ....

....Ran( of has to be included in a term of the co domain of 1 . Formally, given two sets of terms S and S 0 , let us de ne the relation S v S 0 if any term in S is a subterm of a term in S 0 . Let us call the condition Ran( c ) v Ran( c1 ) the redundancy criterion. It is proved in [2] that for the three de ned simpli cations (g = d k c 1 ) is redundant if either c is equivalent to T , or Ran( c ) v Ran( c1 ) To check that a constrained equality is redundant, two special situations may occur. If the rewrite rule has a constraint equivalent to T , redundancy is always ....

H. Kirchner and P-E. Moreau. Prototyping completion with constraints using computational systems. Technical Report 94-R-201, CRIN, 1994.

....formal specification of Input Output for ELAN [27] Proving: ELAN was used in order to implement a predicate prover based on the rules proposed by J. R. Abrial, and implemented in the B tools [Abr96] We developed also a propositional sequent calculus, completion procedures for rewrite systems [20], sufficient conditions for the termination problem [17] A library for automata construction and manipulation has been designed. Approximation automata are used to check conditions for reachability, sufficient completeness, absence of conflicts in systems described by non conditional rewrite ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany), volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

....of medium size developments using ELAN ELAN has been used in many different situations: let us mention some of them. ffl The specification of the unification algorithm for higher order unification based on explicit substitution [Bor95] ffl The specification of constraint completion algorithms [KM95]. ffl The specification of constraint processings [Cas96] ffl The combination of unification algorithms [Rin96] ELAN also been used for specifying disunification, term orderings, a simple Constraint Logic Programming language, narrowing and constraint narrowing. Several examples are presented ....

....normalisation. BKK96a] A description of the operational as well as denotational semantics of the new ELAN strategies, available with version 2.0. This allows the user to define its own strategies as a computational system. KM96] Describes the reflective power of ELAN and of rewriting logic. [KM95] Presents the implementation in ELAN of constraint completion. May 9, 1997 ELAN user manual 66 Contributed works [Bor95] describes the use of explicit substitutions for implementing in ELAN fij normalisation as well as unification of higher order unification. describes the use of ELAN to ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany), volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

....a formal speci cation of Input Output for ELAN [Vir96] Proving: ELAN was used in order to implement a predicate prover based on the rules proposed by J. R. Abrial, and implemented in the B tools [CK97] We developed also a propositional sequent calculus, completion procedures for rewrite systems [KM95], sucient conditions for the termination problem [GG97] A library for automata construction and manipulation has been designed. Approximation automata are used to check conditions for reachability, sucient completeness, absence of con icts in systems described by non conditional rewrite rules ....

....of the general ideas developed in ELAN, with the de nition of computational systems (including the de nition of strategies) and the application of ELAN to design and to prototype constraint programming languages. KM96] An illustration of the re ective power of ELAN and rewriting logic. [KM95] The ELAN implementation of some rule based completion procedures. Kir97] An ELAN tutorial. MK97, MK98, KM98, Mor99] A description of the new compilation techniques developed for ELAN, in particular for associativity and commutativity [Rin97] The use of ELAN to interface various uni cation ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany), volume 914 of Lecture Notes in Computer Science, pages 438-443. Springer-Verlag, 1995.

....using constraints have been designed in Elan, such as a narrowing process with constraints to solve queries in theories defined by rewrite rules, the kernel of a constraint logic programming language using for instance a constraint solver for systems of equations in commutative theories. In [556], a completion process using a basic superposition strategy expressed with constraints, is described in Elan. The strategy language is used for experimenting various simlification strategies in this context. Another significant example developed in Elan is a higher order unification algorithm ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Rewriting Techniques and Applications. 6th International Conference, RTA-95, Kaiserslautern (Germany), pages 438--443. Springer LNCS 914, 1995. \Phi.

....a compiler written in C and Java, a library of standard ELAN modules, a user manual and examples of applications. Among those, let us mention for instance the design of rules and strategies for constraint satisfaction problems [Cas98] theorem proving tools in rst order logic with equality [KM95,CK97] the combination of uni cation algorithms and of decision procedures in various equational theories [Rin97,KR98] More information on the system can be found on the WEB site 1 . A rst ELAN compiler was designed and presented in [Vit96] Experimentations made clear that a higher level of ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany), volume 914 of Lecture Notes in Computer Science, pages 438-443. SpringerVerlag, 1995.

....equalities, one can prove that, if a constrained equality of the form (l = r k oe) is RE1 reducible in the range of oe, it is redundant. More work is needed to obtain satisfactory criteria for elimination of redundancies, but this problem is not addressed here. Further hints can be found in [4, 26, 27, 23, 21]. Definition 4 A derivation from a set of constrained equalities E 0 is a sequence E 0 ; E 1 ; of sets of constrained equalities obtained by successive applications of superposition and eliminations of redundant constrained equalities. Formally: E i = E i Gamma1 [ f(l = r k c)g if (l = r ....

....satisfied, simplification steps can nevertheless be applied if (L 1 k oe 1 ) is transformed by propagation into a new constrained literal (L 1 oe 0 1 k oe 00 1 ) where oe 1 = oe 0 1 oe 00 1 and the condition on ranges is satisfied for oe 00 1 . For more details, see [4] 33] and [21]. 4.4 Completeness results Let INF be the set of inference rules and simplification rules described in Sections 4.2 and 4.3. A set of clauses S is AC unsatisfiable if it has no AC consistent model, i.e. no model consistent with the AC axioms. A contradiction is witnessed by the empty constrained ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. Technical report submitted, 1994.

....us to have an executable description of the deduction process which is the same as the rule based one commonly used in papers. Furthermore, the flexible strategy description allowed by ELAN gives the implementer the possibility to experiment with various completion approaches as described in [KM95] The basis for efficient implementations of computational systems is conditional rewriting controlled by strategies. These strategies are in general nondeterministic, so that the reduction of a term could lead to a family of terms. An efficient compiler, able to manage strategies and this ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany) , volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

....which is the same as the rule based one commonly used in papers. Furthermore, the flexible strategy description allowed by ELAN gives to the implementer the possibility to experiment various completion approaches. Figure 5 gives an idea of the way we encoded it in ELAN. This is described in [KM95] P. Borovansk y, C. Kirchner, H. Kirchner, P. E. Moreau, M. Vittek module constraintCompletion[vars,fss,prec,syst] rules for constraintEquation declare s,t,l,r,g,d : term ; omega : list [int] newConstraint : constraint; newConstraintEq : constraintEquation ; c,c1 : constraint; sigma ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany), volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

....Completion procedures, as many computational processes, can be formulated as instances of a schema that consists of applying rewrite rules on formulas with some strategy, until getting specific normal forms. In this sense they can be understood as computational systems. We have described in [KM95] completion algorithms in ELAN, in which rules to be completed are represented by terms, and the mechanism of simplification (i.e. rewriting steps) is described in ELAN. In a reflective extension, we could transform terms, representing rules, into rewrite rules of the rewriting machine and use ....

H. Kirchner and P.-E. Moreau. Prototyping completion with constraints using computational systems. In J. Hsiang, editor, Proceedings 6th Conference on Rewriting Techniques and Applications, Kaiserslautern (Germany), volume 914 of Lecture Notes in Computer Science, pages 438--443. Springer-Verlag, 1995.

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