Constraint solving for interpolation

Abstract

Interpolation is an important component of recent methods for program verification. It provides a natural and effective means for computing separation between the sets of 'good' and 'bad' states. The existing algorithms for interpolant generation are proof-based: They require explicit construction of proofs, from which interpolants can be computed. Construction of such proofs is a difficult task. We propose an algorithm for the generation of interpolants for the combined theory of linear arithmetic and uninterpreted function symbols that does not require a priori constructed proofs to derive interpolants. It uses a reduction of the problem to constraint solving in linear arithmetic, which allows application of existing highly optimized Linear Programming solvers in black-box fashion. We provide experimental evidence of the practical applicability of our algorithm.

Symbolic optimization with SMT solvers

“…Interpolants are generated from unsatisfiability proofs. For linear real arithmetic, this can be performed by constructing a system of constraints whose solution is a proof of unsatisfiability (and an interpolant) [53]. As recently shown [2], the simplicity of the proof can be crucial to discovering the predicates required for a safe inductive invariant.…”

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

“…In particular, McMillan presents an interpolating theorem prover for rational arithmetic and uninterpreted functions [10]; an interpolating SMT solver for the same logic has been developed by Beyer et al [1]. Rybalchenko et al [16] introduce an interpolation procedure for this logic that works without constructing proofs.…”

“…In order to support expressive programming languages, much effort has been invested in the design of algorithms that compute interpolants for formulae of various theories. As a result, efficient interpolation methods are known for propositional logic, linear arithmetic over the reals with uninterpreted functions [10,1,16], datastructures like arrays and sets [7], and other theories. As for integer arithmetic, a theory particularly relevant for software, interpolating solvers have so far been reported only for restricted fragments such as difference-bound logic, and logics with linear equalities and constant-divisibility predicates.…”

On Interpolation and Symbol Elimination in Theory Extensions

“…In [30] we identify classes of local theory extensions in which interpolants can be computed hierarchically, using a method of computing interpolants in the base theory. [27] proposes an algorithm for the generation of interpolants for linear arithmetic with uninterpreted function symbols which reduces the problem to constraint solving in linear arithmetic. In both cases, when considering theory extensions T 0 ⊆ T 0 ∪ K we devise ways of "separating" the instances of axioms in K and of the congruence axioms.…”