Superposition with completely built-in Abelian groups

Godoy, Guillem; Nieuwenhuis, Robert

doi:10.1016/s0747-7171(03)00070-1

Cited by 8 publications

(26 citation statements)

References 25 publications

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“…In [12] the authors give a superposition calculus in which the reasoning about elements of an abelian group is completely built-in. Our aim is to elaborate that calculus so that it provides a decision procedure for the satisfiability problem modulo theories modelling interesting data structures and extending AG.…”

Section: A Calculus For Abelian Groupsmentioning

confidence: 99%

“…First of all, we will reason over terms modulo an AG-rewriting system: quoting [12], the system R AG consists of the rules (i)…”

Section: A Calculus For Abelian Groupsmentioning

confidence: 99%

“…Now, we are ready to describe the calculus. We will rely basically on three rules, Direct AG-superposition, Inverse AG-superposition and Reflection, and, as in [12], we will apply the rules only in case the premises satisfy certain conditions as explained in the following. Moreover, from now on we assume that all the literals will be eagerly maintained in AG-normal form, meaning that they will be maintained as (dis)equations between terms in AG-normal form.…”

Section: A Calculus For Abelian Groupsmentioning

confidence: 99%

“…The case of Associativity-Commutativity has been extensively investigated since it appears in many equational theories, and among them, the theory of abelian groups is a very good candidate as fragment of arithmetic. Recently, the standard superposition calculus [19] has been extended to a superposition calculus modulo the built-in theory of abelian groups [12]. This work paves the way for the application of a superposition calculus modulo a fragment of arithmetic to build decision procedures of practical interest in verification.…”

Section: Introductionmentioning

confidence: 99%

“…the satisfiability of constraints. Second, we identify a class of theories that extend the theory of abelian groups and for which a simplified constraint-free (but many-sorted) version of the superposition calculus introduced in [12] is proved to be complete. This superposition calculus allows us to obtain effective decision procedures that can be plugged into the non-disjoint extension of the Nelson-Oppen combination method.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Combinable Extensions of Abelian Groups

Nicolini

Ringeissen

Rusinowitch

2009

Automated Deduction – CADE-22

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Abstract:The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination methodà la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this nondisjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification.Key-words: Satisfiability Procedure, Combination, Equational Reasoning, Union of Non-Disjoint Theories, Arithmetic, Abelian Groups * E-mail: FirstName.LastName@loria.fr Extensions combinables des groupes abéliensRésumé : La conception de procédures de décision pour la combinaison de théories partageant un fragment d'arithmétique est un défi dans le domaine de la vérification. Une solution possible consisteà appliquer une méthode de combinaisonà la Nelson-Oppen, comme celle développée par Ghilardi pour l'union de théories non-disjointes. On montre comment appliquer cette méthode de combinaison non-disjointe avec la théorie des groupes abéliens comme théorie partagée. On considère la complétude et l'effectivité de cette méthode. Pour la complétude, on montre que la théorie des groupes abéliens peut se plonger dans une théorie admettant l'élimination des quantificateurs. Pourêtre effectif, on utilise un calcul de superposition modulo la théorie des groupes abéliens qui est montré complet pour des théories intéressantes en pratique dans le domaine de la vérification.

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Section: A Calculus For Abelian Groupsmentioning

confidence: 99%

“…First of all, we will reason over terms modulo an AG-rewriting system: quoting [12], the system R AG consists of the rules (i)…”

Section: A Calculus For Abelian Groupsmentioning

confidence: 99%

Section: A Calculus For Abelian Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%