An Instantiation Scheme for Satisfiability Modulo Theories

Abstract

International audienceState-of-the-art theory solvers generally rely on an instantiation of the axioms of the theory, and depending on the solvers, this instantiation is more or less explicit. This paper introduces a generic instantiation scheme for solving SMT problems, along with syntactic criteria to identify the classes of clauses for which it is complete. The instantiation scheme itself is simple to implement, and we have produced an implementation of the syntactic criteria that guarantee a given set of clauses can be safely instantiated. We used our implementation to test the completeness of our scheme for several theories of interest in the SMT community, some of which are listed in the last section of this paper

Modular instantiation schemes

“…instantiation procedures are available in the literature [9,6,7,1,3,5]. Our goal in this paper is to provide a general mechanism for constructing new complete instantiation procedures by combining existing ones.…”

“…As an example of application, consider the specification T Z defined in Section 1 and any so-called controllable specification T (in the sense of [5], e.g. the theory of arrays).…”

“…The scheme is obviously base-complete (since the considered instances do not depend on the ground clauses). Furthermore, in [5] a terminating and base-complete instantiation scheme Θ is defined for controllable specifications. By Theorem 6, Θ Z Θ is a complete instantiation scheme for T Z ∪ T .…”

“…Any instantiation scheme can be viewed as a function Θ that maps a set of clauses S to a set of ground instances Θ(S) of clauses in S; such a scheme is refutationally complete if for all sets of clauses S, Θ(S) is satisfiable exactly when S is. Examples of refutationally complete instantiation schemes include [9,6,7]. In general, Θ(S) may be infinite; however, terminating and refutationally complete instantiation schemes have been defined for specific theories [1,3,5].…”

“…Examples of refutationally complete instantiation schemes include [9,6,7]. In general, Θ(S) may be infinite; however, terminating and refutationally complete instantiation schemes have been defined for specific theories [1,3,5]. Such decision procedures can be very attractive, provided they do not generate too many ground clauses, since it is much easier to test the satisfiability of a ground set of clauses than that of a non-ground set [2,4].…”