Combinable Extensions of Abelian Groups

Abstract

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.

Data Structures with Arithmetic Constraints: A Non-disjoint Combination

“…In [17], the shared theory is a more precise approximation of the theory of integers, but on the other hand there is no integration of standard techniques for reasoning about arithmetic. In [16], we show how to combine data structures sharing the theory of abelian groups. In a similar way to what is investigated here, it would be interesting to study the combination of a data structure with some fragments of arithmetic when the shared theory is the one of abelian groups.…”

A Rewriting Approach to the Combination of Data Structures with Bridging Theories

“…This problem was first studied for disjoint combinations, but nondisjoint unions naturally arise when considering a bridging theory to relate the data structure theory T 1 to the arithmetic theory T 2 , e.g. the length function for the data structure of lists [13,14,16]. The Ghilardi non-disjoint combination method [11] has been already applied to handle some connections between theories [3,13,14].…”

“…the length function for the data structure of lists [13,14,16]. The Ghilardi non-disjoint combination method [11] has been already applied to handle some connections between theories [3,13,14]. In [13,14], the idea is to use superposition-based satisfiability procedures to process theory extensions of T 2 .…”

“…The Ghilardi non-disjoint combination method [11] has been already applied to handle some connections between theories [3,13,14]. In [13,14], the idea is to use superposition-based satisfiability procedures to process theory extensions of T 2 . In that context, it is always a difficult and tedious task to design a new superposition calculus incorporating T 2 as a built-in theory.…”

Modular Termination and Combinability for Superposition Modulo Counter Arithmetic

“…This is especially the case when we consider theories sharing some algebraic constraints [14,16,17,18,20,21]. In order to combine satisfiability procedures for the single theories to handle constraints in their nondisjoint union one needs to rely on powerful methods such as the combination framework of [9,10].…”