Satisfiability Procedures for Combination of Theories Sharing Integer Offsets

Abstract

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.

Combinable Extensions of Abelian Groups

“…As a first direction, we would like to relax current restrictions on theories and saturation types to apply effectively the calculus in the non-disjoint combination method. At the moment, since the presence of variables of sort ag into the clauses is not allowed, the results in [18] are not subsumed by the present paper. That restriction is justified by technical reasons: an important issue would be to discard it, enlarging in this way the applicability of our results.…”

“…[18]), we can substitute T L with the set of the purely equational axioms of T L , say T L ′ , and enrich a bit the set of literals G to a set of literals G ′ in such a way…”

“…We have (see e.g. [18]) that G is satisfiable w.r.t. T L ∪ T ℓ ∪ AG if and only if G ′ is satisfiable w.r.t.…”

“…An extension of the Nelson-Oppen combination method to the non-disjoint case has been proposed in [11]. This non-disjoint combination framework has been recently applied to the theory of Integer Offsets [18]. In this paper, our aim is to consider a more expressive fragment by studying the case of abelian groups.…”

Data Structures with Arithmetic Constraints: A Non-disjoint Combination

“…In this paper, we show how to use this non-disjoint combination method to build a decision procedure for the union of (1) a (convex) theory modeling some data structure and whose successor function that expresses some counting capabilities, and (2) the linear or non-linear arithmetic over the rationals augmented by the successor function. Both theories share the successor function s and have a common subtheory, called the theory of Increment, axiomatizing the acyclicity and the injectivity of s. This paper is the continuation of a previous work, where we studied the combination of superposition-based decision procedures for the union of two data structures sharing the theory of Integer Offsets [17]. Unfortunately, it was not possible in [17] to integrate standard procedures for reasoning about arithmetic.…”