On Variable-inactivity and Polynomial Formula-Satisfiability Procedures

Frontiers of Combining Systems

Senni

2011

Abstract. Modularity is a highly desirable property in the development of satisfiability procedures. In this paper we are interested in using a dedicated superposition calculus to develop satisfiability procedures for (unions of) theories sharing counter arithmetic. In the first place, we are concerned with the termination of this calculus for theories representing data structures and their extensions. To this purpose, we prove a modularity result for termination which allows us to use our superposition calculus as a satisfiability procedure for combinations of data structures. In addition, we present a general combinability result that permits us to use our satisfiability procedures into a non-disjoint combination method a la Nelson-Oppen without loss of completeness. This latter result is useful whenever data structures are combined with theories for which superposition is not applicable, like theories of arithmetic.

Section: Superposition Calculus For Counter Arithmeticmentioning

confidence: 99%

“…As an alternative, we propose a modular termination result that applies to the union of the considered theories and is based on the analysis of the saturations (similarly to [1,4]). This result is interesting in that it does not require us to prove the more complex property of T C -compatibility for the component theories.…”

Section: Modular Terminationmentioning

confidence: 99%

Modular Termination and Combinability for Superposition Modulo Counter Arithmetic

Frontiers of Combining Systems

Senni

2011

“…Recent literature has focused on the possibility of using the superposition calculus in order to decide the satisfiability of ground formulae modulo the theory of Integer Offsets and some disjoint extensions [1,3]. Contrary to those papers, we are interested in a superposition-based calculus to deal with non-disjoint extensions of Integer Offsets, being able to constraint the successor symbol with additional axioms.…”

Section: Superposition Calculus For Integer Offsetsmentioning

confidence: 99%

“…Second, rewriting techniques are of prime interest to design satisfiability procedures which can be efficiently plugged into the disjoint combination framework [11]. In some particular cases, the rewriting approach is an alternative to the combination approach by allowing us to build superposition-based satisfiability procedures for combinations of finitely axiomatized theories, including the theory of Integer Offsets [1,3], but these theories must be over disjoint signatures.…”

Section: Introductionmentioning

confidence: 99%

Satisfiability Procedures for Combination of Theories Sharing Integer Offsets

Nicolini

Tools and Algorithms for the Construction and Analysis of Systems

Rusinowitch

2009

Abstract. We present a novel technique to combine satisfiability procedures for theories that model some data-structures and that share the integer offsets. This procedure extends the Nelson-Oppen approach to a family of non-disjoint theories that have practical interest in verification. The result is derived by showing that the considered theories satisfy the hypotheses of a general result on non-disjoint combination. In particular, the capability of computing logical consequences over the shared signature is ensured in a non trivial way by devising a suitable complete superposition calculus.

“…To improve the applicability of SMT solvers, it is important to develop general uniform methods to combine and to build decision procedures. Hence, equational theorem proving has been successfully applied to build decision procedures for various data structures including lists and arrays [2,1,4,13,6]. More precisely, a superposition calculus [18] based on rewriting techniques can be used for this purpose.…”

Section: Introductionmentioning

confidence: 99%

Data Structures with Arithmetic Constraints: A Non-disjoint Combination

Nicolini

Frontiers of Combining Systems

Rusinowitch

2009

Abstract:We apply an extension of the Nelson-Oppen combination method to develop a decision procedure for the non-disjoint union of theories modeling data structures with a counting operator and fragments of arithmetic. We present some data structures and some fragments of arithmetic for which the combination method is complete and effective. To achieve effectiveness, the combination method relies on particular procedures to compute sets that are representative of all the consequences over the shared theory. We show how to compute these sets by using a superposition calculus for the theories of the considered data structures and various solving and reduction techniques for the fragments of arithmetic we are interested in, including Gauss elimination, Fourier-Motzkin elimination and Groebner bases computation.