From Strong Amalgamability to Modularity of Quantifier-Free Interpolation

Abstract

The use of interpolants in verification is gaining more and more importance. Since theories used in applications are usually obtained as (disjoint) combinations of simpler theories, it is important to modularly re-use interpolation algorithms for the component theories. We show that a sufficient and necessary condition to do this for quantifierfree interpolation is that the component theories have the 'strong (sub-)amalgamation'property. Then, we provide an equivalent syntactic characterization, identify a sufficient condition, and design a combined quantifier-free interpolation algorithm capable of handling both convex and non-convex theories, that subsumes and extends most existing work on combined interpolation.

Complete instantiation-based interpolation

“…Our notion of partial amalgamation is closely related to the (strong) amalgamation property [32], whose role in ground interpolation for disjoint theory combinations has been recently studied [11]. Our use of amalgamation properties is orthogonal to [11], as we consider (non-disjoint) theory extensions rather than disjoint theory combinations.…”

“…Our use of amalgamation properties is orthogonal to [11], as we consider (non-disjoint) theory extensions rather than disjoint theory combinations. In a sense, partial amalgamation is the adaptation of the weak embedability condition in [46] to the case of interpolation.…”

“…In a sense, partial amalgamation is the adaptation of the weak embedability condition in [46] to the case of interpolation. Our approach can thus be thought of as the symbioses of the two orthogonal approaches described in [46,47] and [11]. Note that neither of the interpolation techniques presented in [47] and [11] can be applied directly to the theory of lists considered in this paper.…”

“…Our approach can thus be thought of as the symbioses of the two orthogonal approaches described in [46,47] and [11]. Note that neither of the interpolation techniques presented in [47] and [11] can be applied directly to the theory of lists considered in this paper. The approach in [47] is restricted to extension axioms of a very specific syntactic form: Horn clauses in which all predicate symbols are binary and where additional guard constraints on the quantified variables apply.…”

“…Modern SMT solvers implement ground interpolation procedures for the theories that are most commonly used in program verification. This includes theories such as linear arithmetic [8,9,23,38], the theory of uninterpreted function symbols with equality [19,38,50], and combinations of such theories [11,21,38,50]. However, many application-specific theories remain unsupported.…”