Laurent Vigneron. Associative-commutative deduction with constraints. In Proc. 12th Int. Conf. on Automated Deduction, LNCS 814, pages 530-544, Nancy, France, 1994. Springer.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... recognized that automated theorem provers have problems proving theorems in theories with permutative axioms like associativity, commutativity, distributivity and the inverse law which are common in algebra, and there have been various approaches to the integration of these axioms into provers [17, 11, 16, 5, 2, 8, 13, 14, 15]. A similar argument holds for the use of equivalences on the level of logical formulas. State of the art resolution theorem provers such as SPASS do a clause normal form transformation once at the beginning, which destroys in particular the equivalences. With some e ort it is possible to ....

Laurent Vigneron. Associative-commutative deduction with constraints. In Proc. 12th Int. Conf. on Automated Deduction, LNCS 814, pages 530-544, Nancy, France, 1994. Springer.

....that changes is the interpretation of the constraints: now = is interpreted as E equality, instead of syntactic equality 2 . Apart from the basicness restriction, an additional advantage is now that only one conclusion is generated per inference, instead of one conclusion for each E unifier[Vig94,NR97]. This can have dramatic consequences. For example, there are more than a million unifiers in mguAC (f(x; x; x) f(y 1 ; y 2 ; y 3 ; y 4 ) 2.4 Redundancy and saturation Roughly, a clause C is redundant in a set of clauses S if C is a logical consequence of smaller (with respect to the given ....

Laurent Vigneron. Associative Commutative Deduction with constraints. In Alan Bundy, editor, 12th International Conference on Automated Deduction (CADE), LNAI 814, pages 530--544, Nancy, France, June 1994. SpringerVerlag.

.... signatures) were given in [Nie93,NRV99] For the Knuth Bendix ordering (KBO) this result has only recently been obtained (for fixed signatures) in [KV00] Ordered strategies and ordering constraint inheritance can be used without loosing completeness with built in algebraic theories E, like AC [NR97,Vig94] or AG [GN00] An additional advantage of constraints in this context is that in each inference only one conclusion is generated, instead of one conclusion for each Eunifier. This can have dramatic consequences. For example, there are more than a million unifiers in mguAC (f(x; x; x) f(y 1 ; y 2 ....

Laurent Vigneron. Associative Commutative Deduction with constraints. In Allan Bundy, editor, 12th International Conference on Automated Deduction (CADE), LNAI 814, pages 530--544, Nancy, France, June 1994. SpringerVerlag.

.... recognized that automated theorem provers have problems proving theorems in theories with permutative axioms like associativity, commutativity, distributivity and the inverse law which are common in algebra, and there have been various approaches to the integration of these axioms into provers [17, 11, 16, 5, 2, 8, 13, 14, 15]. A similar argument holds for the use of equivalences on the level of logical formulas. State of the art resolution theorem provers such as SPASS do a clause normal form transformation once at the beginning, which completely destroys in particular the equivalences. With some e ort it is possible ....

Laurent Vigneron. Associative-commutative deduction with constraints. In Proc. 12th Int. Conf. on Automated Deduction, LNCS 814, pages 530-544, Nancy, France, 1994. Springer.

.... progresses on the working group topics Combination problems for logics and constraints of particular interest Our research in this area has always been closely related to the progress made by our group and other CCL partners on completeness of deduction methods dealing with constrained clauses [72, 71, 100]. By expressing particular strategies at the formula level, these methods make it possible to importantly cut down the search space by inheriting additional (ordering and unifyability) restrictions among the clauses. In particular, during this year we have finally succeeded in our purpose of ....

.... algorithms) the first idea is to guess at once all identical subterms, an idea which is also applied (and extended because also the ordering relations are guessed) in [30] and [69] Theorem proving and constraints In [70] it is shown that the methods for deduction with constrained clauses [72, 71, 100] turn out to be applicable as well to answer computation (the computation of values for existentially quantified varibles, as done in logic programming, automated theorem proving discovering or deductive data bases) By dealing with narrowing and refutational theorem proving in a uniform way, the ....

Laurent Vigneron. Associative Commutative Deduction with constraints. In Allan Bundy, editor, 12th International Conference on Automated Deduction, LNAI, Nancy, France, June 1994. Springer-Verlag.

....Simplify CE [ fp = q k cg 7 7 CE [ fp 0 = q k cg if p (g=d k T) oe CE;AC;L p 0 and p oe(g) or q oe(d) Fig. 5. CCM: Constrained Completion Modulo AC A refutationally complete theorem prover based on constrained paramodulation and using a different proof technique is proposed in [51]. An additional difficulty is to prove that completeness is preserved when simplification and deletion are incorporated into constrained completion. This is only true under some conditions, proposed for the empty theory in [41] and for associative and commutative theories in [51] Indeed, a ....

....is proposed in [51] An additional difficulty is to prove that completeness is preserved when simplification and deletion are incorporated into constrained completion. This is only true under some conditions, proposed for the empty theory in [41] and for associative and commutative theories in [51]. Indeed, a simple sufficient condition is to allow simplification only by unconstrained rewrite rules, as expressed in the condition of rule Simplify in Figure 5. A restricted form of simplification is applied in this process and the next section is devoted to a more powerful notion of ....

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L. Vigneron. Associative-commutative deduction with constraints. In A. Bundy, editor, Proceedings of CADE-12, volume 814 of Lecture Notes in Computer Science, pages 530--544. Springer-Verlag, 1994.

....: constrained and basic completion, equational theory, completion modulo, simpli cation. 1 Introduction This paper takes its root from the study of simpli cation in the case of Basic Completion [NR92a,BGLS95,LS98] and especially, in the case of Basic Completion modulo an equational theory E [Vig94,NR94]. Basic Completion is a restriction of Knuth Bendix Completion [KB70] where constraints are used. We also call Knuth Bendix Completion Standard Completion. Deduction with constraints was introduced in [KKR90] In Basic Completion, the use of equational constraints permits us to save the most ....

....uses the advantages of Basic Completion where AC uni cation constraints are saved. This permits us to have only one deduced equation instead of a doubly exponential number when constraints are not used. Basic Simpli cation is the only known complete method for Basic Completion modulo the AC theory [Vig94]. In this method, the complete set of solutions of the AC constraints of the premises must be computed knowing that the cardinality of this set is doubly exponential. Then, AC matching problems are computed taking into account the solutions of the AC uni cation problem. That is why Basic Simpli ....

L. Vigneron. Associative-Commutative Deduction with Constraints. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Articial Intelligence, pages 530-544. Springer-Verlag, June 1994.

....every finite E. The completeness of such a fully basic strategy for the AC case (combined with ordering constraints) was first proved in [Nieuwenhuis and Rubio 1994, Nieuwenhuis and Rubio 1997] although the first results on (almost basic) constrained deduction methods modulo AC were reported in [Vigneron 1994]. The basicness restriction is considered to have been a key strategy by McCune [1997b] in his celebrated ACparamodulation based proof of the Robbins problem. In Section 6.3 of this chapter basic paramodulation modulo AC is explained. Paramodulation based theorem proving 11 2. Preliminaries ....

....the useful extended clauses can be easily characterized. Then it becomes possible as well to design specific inference rules instead of handling these extensions explicitly. This is the way most paramodulation calculi for the AC case are expressed [Paul 1992, Rusinowitch and Vigneron 1995, Vigneron 1994, Nieuwenhuis and Rubio 1997] and in Section 6.2 (see also [Rubio 1996] for builtin semigroups, i.e. associative theories) This approach is considered as well for arbitrary regular theories in [Vigneron 1996] Recent research concerns algebraic structures richer than abelian semigroups, like ....

Vigneron L. [1994], Associative Commutative Deduction with constraints, in A. Bundy, ed., `12th International Conference on Automated Deduction (CADE)', LNAI 814, SpringerVerlag, Nancy, France, pp. 530--544.

....Instead, the unification problems are stored in the constraints and a constrained clause C j T is redundant if T is unsatisfiable. Apart from the well known basicness restriction, an additional advantage is that only one conclusion is generated, instead of one conclusion for each AG unifier[20, 16]. Checking the ordering restrictions in our framework is different from the usual situation. Instead of checking whether, say, for given terms s and t, there exists some ground oe such that soe rpo toe, we need to check whether this holds after normalising both sides by RAG , that is, whether ....

L. Vigneron. Associative Commutative Deduction with constraints. In A. Bundy, editor, 12th International Conference on Automated Deduction (CADE), LNAI 814, pages 530-- 544, Nancy, France, June 1994. Springer-Verlag.

....Instead, the unification problems are stored in the constraints and a constrained clause C j T is redundant if T is unsatisfiable. Apart from the well known basicness restriction, an additional advantage is that only one conclusion is generated, instead of one conclusion for each AG unifier[20, 16]. Checking the ordering restrictions in our framework is different from the usual situation. Instead of checking whether, say, for given terms s and t, there exists some ground oe such that soe rpo toe, we need to check whether this holds after normalising both sides by RAG , that is, whether ....

L. Vigneron. Associative Commutative Deduction with constraints. In A. Bundy, editor, 12th International Conference on Automated Deduction (CADE), LNAI 814, pages 530-- 544, Nancy, France, June 1994. Springer-Verlag.

....The only aspect that changes is the interpretation of the constraints: now = is interpreted as E equality, instead of syntactic equality 2 . Apart from the basicness restriction, an additional advantage is now that only one conclusion is generated, instead of one conclusion for each Eunifier [Vig94,NR97]. This can have dramatic consequences. For example, there are more than a million unifiers in mguAC (f(x; x; x) f(y 1 ; y 2 ; y 3 ; y 4 ) 2 Although for some E extended inference rules are needed (see, e.g. RV95] 2.4 Redundancy and saturation Roughly, a clause C is redundant in a set of ....

Laurent Vigneron. Associative Commutative Deduction with constraints. In Allan Bundy, editor, 12th International Conference on Automated Deduction, LNAI 814, pages 530--544, Nancy, France, June 1994. Springer-Verlag.

....the A unifiability of T to know whether an inconsistency has been derived or not. By means of the model generation This work has been partially supported by the Esprit Working Group CCL, ref. 6028 method [3] we proved the completeness of such a basic strategy for the AC case in [12] see also [17]) combined with ordering constraints. Here we develop a similar inference system for the case modulo A, together with an essential ingredient for these methods, namely the first as far as we know A compatible reduction ordering total on the ground A congruence classes. This paper is ....

Laurent Vigneron. Associative Commutative Deduction with constraints. In Allan Bundy, editor, 12th International Conference on Automated Deduction, LNAI, Nancy, France, June 1994. Springer-Verlag. 13 This article was processed using the L a T E X macro package with LLNCS style 14

....constraints. E cycles may only occur when an equation simplies an ancestor. Whenever a simplication would create an E cycle in the dependency graph, we disallow the simplication. There are several methods of proof of completeness. One of these methods consists in using semantics trees [Vig94]. Our completeness proof is based on the model construction proof of [BG94] which is also used in the completeness proofs of Basic Completion in [BGLS95, NR92] Like those proofs, we build a model of irreducible equations, based on an ordering of the equations. The dioeerence is that we do not ....

L. Vigneron. Associative-Commutative Deduction with Constraints. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Articial Intelligence, pages 530544. Springer-Verlag, June 1994.

....different approach to integer modules. The problems are restricted to Horn clauses, that is deducing one equation from a set of equations. Completeness is shown only for the case without uninterpreted function symbols. Wertz (1992) Bachmair and Ganzinger (1994a) Nieuwenhuis and Rubio (1994) and Vigneron (1994) consider superposition calculi modulo AC, and the last three also use constraints. 13 Conclusion and Further Work We have presented a refutationally complete superposition calculus for first order theories that contain abelian groups or integer modules. We have also shown that certain variables ....

Vigneron, L. (1994). Associative-commutative deduction with constraints. In Proc. 12th Int. Conf. on Automated Deduction, Nancy, France, LNCS 814, pp. 530--544. Springer.

.... (AC) function symbols, instead of AC unifying the terms and generating a double exponential number of new clauses, the equality can be kept in an equality constraint (and then interpreted in an AC theory) Completeness of the corresponding strategies was shown concurrently in [721] and in [916, 917]. In [822] a complete analysis of these constrained deduction methods is presented, also covering the case modulo A [826] and analysing other methods for deduction modulo equational theories E when E unification is infinitary of even undecidable. In [718] it is shown that these ideas turn out ....

L. Vigneron. Associative Commutative Deduction with constraints. In A. Bundy, editor, 12th International Conference on Automated Deduction, Nancy, France, June 1994. Springer-Verlag LNAI.

....new proof technique given here for building in AC is interesting in itself (apart from the advantages due to the symbolic constraints) because of its simplicity. The first results on (almost basic) constrained deduction modulo AC were reported by Laurent Vigneron. In a recent version of his work (Vigneron, 1994) , the computation 4 R. Nieuwenhuis and A. Rubio of AC unifiers is also avoided (by applying our notion of irreducibility, defin. 4.3) and some further refinements as well as examples by an implementation of these techniques are given. His proof methods are completely different from ours and ....

....like redex orderings and variable abstraction (Bachmair et al. 1995) can also be smoothly incorporated here. Regarding simplification and other redundancy methods, the abstract redundancy notions given here express sharp bounds on the existing concrete redundancy methods (like the ones given in Vigneron, 1994 , which indeed fit into our abstract ones) 2. Basic notions and terminology An equation is a multiset of terms fs; tg, which will be written s t. A first order clause is a pair of finite multisets of equations Gamma (the antecedent) and Delta (the succedent) denoted by Gamma Delta. By ....

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Vigneron, L. (1994). Associative Commutative Deduction with constraints. In Allan Bundy, editor, 12th International Conference on Automated Deduction, LNAI 814, pages 530--544, Nancy, France.

....in the presence of associative and commutative (AC) function symbols. This yields an algorithm for solving AC RPO constraints (where ACRPO is the AC compatible total reduction ordering of [16] which was a missing ingredient for automated deduction strategies with AC constraint inheritance [15, 19]. As in the AC unification case (actually the AC unification algorithm of [9] is an instance of ours) for this purpose we first study the pure case, i.e. we show how to solve AC ordering constraints built over a single AC function symbol and variables. Since AC RPO is an interpretation based ....

....recursive path orderings (RPO) introduced by N. Dershowitz [6] The existential fragment of the theory of each RPO was shown decidable [4, 8] and actually NP complete [13] Ordered strategies and ordering constraints inheritance can be used in presence of AC symbols without loosing completeness [15, 19]. This requires first an ordering which, in addition to former requirements, is also AC compatible. Then we need a constraint solving algorithm for such an ordering. AC compatible (monotonic, well founded, total on ground equivalence classes) orderings were unknown until recently, when P. ....

L. Vigneron. Associative Commutative Deduction with constraints. In A. Bundy, editor, 12th International Conference on Automated Deduction, LNAI, Nancy, France, June 1994. SpringerVerlag.

....designed by delaying complex problem solving and pruning some parts of the search tree. Constraints also schematize (infinitely) many objects, especially substitutions or ground instances of equalities. Recent work involves irreducibility constraints [7] and deduction processes with AC constraints [12, 13]. However introducing equational and ordering constraints in completion processes needs a cautious analysis. For efficiency reasons, simplification is essential in completion but more difficult in the context of constrained equalities. As solutions are not computed, a constrained equality is ....

....the simplified formula has to be checked. A first motivation for this work was to design a completion process with constraints that gives priority to simplifications, and to compare it with completion without constraints. We relied on previous theoretical results described in other papers, mainly [10, 12, 13] to elaborate a precise notion of simplification with constraints and check redundancy of the simplified formula. In order to clarify the combination of weakening by propagation and simplification, we wanted to perform experimentations with different strategies for simplification and ....

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L. Vigneron. Associative-Commutative Deduction with Constraints. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 530--544. Springer-Verlag, June 1994.

....T . If C is the empty clause one can decide the AC unifiability of T (which is NP complete, cf. KN92] to know whether an inconsistency has been derived or not. The first results on (almost basic) constrained deduction modulo AC were reported by Laurent Vigneron. In a recent version of his work [Vig94] he also avoids the computation of AC unifiers (by applying our notion of irreducibility, defin. 3.3) and defines several additional restrictions. His proofs are completely different from ours and based on transfinite semantic trees as in [RV93] He also reports on an implementation of these ....

....S with C D. Then RC RD , ACC ACD , and I C I D . 1 Here point 2. is based on the irreducibility notion of [BGLS92] and point 3. is the crucial trick in our proof for lifting in the AC case and thus avoiding the computation of AC unifiers, cf. example 3.13. Vigneron applies our idea in [Vig94] and extends it by allowing point 3 only if x is a new variable of AC superposition, which permits to impose some more restrictions on the inferences (this also works for the proofs given here) Proof The first point, RC RD , holds by definition. For ACC ACD : suppose some s s 0 2 ACD ....

Laurent Vigneron. Associative Commutative Deduction with constraints. In Allan Bundy, editor, 12th International Conference on Automated Deduction, LNAI, Nancy, France, June 1994. Springer-Verlag.

....needed only at the end if one wants to uncompress such a constraint into its (doubly exponentially many) concrete substitutions. In [NR94] by means of the model generation method, we proved the completeness of such a basic strategy for the AC case combined with ordering constraints (see also [Vig94], and see [Rub94] for a detailed analysis of constrained theorem proving modulo equational theories) Indeed, when working modulo AC and considering the inference system I AC , Theorem 14 holds too. 7 Conclusions and Further Work We have given a uniform completeness proof covering both ....

Laurent Vigneron. Associative Commutative Deduction with constraints. In Allan Bundy, editor, 12th International Conference on Automated Deduction, LNAI, Nancy, France, June 1994. Springer-Verlag.

.... techniques as in [7, 22] Unification algorithms may be solved by term rewriting techniques too, for dealing with parts of these theories such as in [15] However, it seems that one of the most interesting ways for dealing with these problems of E unification is to use symbolic constraints, as in [28]. Acknowlegments: I would like to thank Prof. Anita Wasilewska of Stony Brook for the numerous discussions we had on the history of the bases of this paper. I would like to dedicate this paper to the memory of my colleague Valentin Antimirov of INRIA Lorraine (France) with whom I had very ....

L. Vigneron. Associative-Commutative Deduction with Constraints. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 530--544. Springer-Verlag, June 1994.

....has to be maximal w.r.t. the other negative literals of the clause, ffl Simplification rules: they permit the deletion of redundant clauses. The above extensions were defined for equational theories. For the case of associative and commutative theories (AC) we define a constraint strategy [6]. This extends results of H. C. Kirchner and Rusinowitch, Nieuwenhuis and Rubio, and Bachmair, Ganzinger, Lynch and Snyder. The principle of our constraint strategy is simple: each unification problem is not solved; it is added to a conjunction of constraints c joined to the deduced clause C. So, ....

L. Vigneron. Associative-Commutative Deduction with Constraints. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of LNAI, pages 530--544. Springer-Verlag, June 1994.

....Campus Scientifique Batiment LORIA Boite Postale 239 54506 Vandoeuvre l es Nancy Cedex T el ephone : 33) 83.59.20.00 T el ecopie : 33) 83.41. 30.79 Associative Commutative Deduction with Constraints Laurent Vigneron CRIN 93 R 196 Novembre 1993 Presented at CADE 12, Nancy, June 1994 [Vig94a] Associative Commutative Deduction with Constraints Laurent Vigneron CRIN CNRS INRIA Lorraine BP 239, 54506 Vandoeuvre l es Nancy Cedex, France E mail : Laurent.Vigneron loria.fr Abstract Associative commutative equational reasoning is known to be highly complex for theorem proving. ....

L. Vigneron. Associative-Commutative Deduction with Constraints. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 530--544. Springer-Verlag, June 1994.

....plus large de th eories. Les contraintes symboliques ont et e introduites dans [KKR90] pour exprimer les probl emes d unification et d ordre. Elles g en eralisent la strat egie basique d efinie dans [Hul80] comme cela a et e montr e dans [BGLS92, NR92a, NR92b] pour la th eorie vide et dans [NR94, Vig94] pour les th eories associatives et commutatives, o u des proc edures r efutationellement compl etes, bas ees sur les strat egies de paramodulation et de superposition, ont et e d efinies pour ne jamais avoir a calculer un seul unificateur. Seule la satisfaisabilit e des probl emes ....

....lors d etapes de superposition. Pour plus de d etails, voir [PS81] 2 A propos des r egles de simplification Une d efinition de r egles de simplification compatibles avec des r egles d inf erence comme celles cit ees ci dessus a et e propos ee dans [BGLS92] pour la th eorie vide, et dans [Vig94] pour les th eories associatives et commutatives. Il existe deux r egles principales de simplification : ffl subsomption : etant donn ee une clause C 1 j[ c 1 ]j et une solution oe 1 de c 1 , l instance C 1 j[ oe 1 ]j de C 1 j[ c 1 ]j subsume une clause C 2 j[ c 2 ]j si, pour chaque solution oe ....

[Article contains additional citation context not shown here]

L. Vigneron. Associative-Commutative Deduction with Constraints. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 530--544. Springer-Verlag, June 1994.

....of the memory size of the computer. The implemented techniques involve ordering and simpli cation strategies combined with a deduction system based on paramodulation and resolution, as mentioned earlier, but other important strategies are also available, as the superposition and basic strategies [16]. At the end of an execution, the user can ask for a lot of extra information. Especially the prover can present a proof of a derived property, or of the contradiction found. Many statistics are also available, such as the number of deduction steps, the number of simpli cation steps, and the ....

L. Vigneron. Associative-Commutative Deduction with Constraints. In A.Bundy, ed., Proc. 12th Int. Conf. on Automated Deduction, Nancy (France), vol. 814 of LNAI, pp. 530544. Springer-Verlag, 1994.

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