A Simple and Flexible Way of Computing Small Unsatisfiable Cores in SAT Modulo Theories

Finding small unsatisfiable cores for SAT problems has recently received a lot of interest, mostly for its applications in formal verification. Surprisingly, the same problem in the context of SAT Modulo Theories (SMT) has instead received very little attention in the literature; in particular, we are not aware of any work aiming at producing small unsatisfiable cores in SMT. The purpose of this paper is to start filling the gap in this area, by proposing a novel approach for computing small unsat cores in SMT. The main idea is to combine an SMT solver with an external propositional core extractor: the SMT solver produces the theory lemmas found during the search; the core extractor is then called on the boolean abstraction of the original SMT problem and of the theory lemmas. This results in an unsatisfiable core for the original SMT problem, once the remaining theory lemmas have been removed. The approach has several advantages: it is extremely simple to implement and to update, and it can be interfaced with every propositional core extractor in a plug-and-play way, so that to benefit for free of all unsatcore reduction techniques which have been or will be made available. 1 Motivations and goals In the last decade we have witnessed an impressive advance in the efficiency of SAT techniques, which has brought large and previously intractable problems at the reach of state-of-the-art SAT solvers. In particular, and due to its importance in formal verification, the problem of finding small unsatisfiable cores in SAT — i.e., unsatisfiable subsets of unsatisfiable sets of clauses— has been addressed by many authors in the recent years [8, 9, 11, 4, 7, 10]. The formalism of plain propositional logic, however, is often not suitable or expressive enough for representing many interesting real-world problems, which are more naturally expressible as satisfiability problems in decidable first-order theories —Satisfiability Modulo Theories, SMT. Efficient SMT solvers have been developed in the last five years, called lazy SMT solvers, which combine DPLL with ad-hoc decision procedures for many theories of interest (e.g., [6, 1, 2, 5]). Surprisingly, the problem of finding unsatisfiable cores in SMT has received virtually no attention in the literature. Although some SMT tools do compute unsat cores, this is done either as a byproduct of the more general task of producing proofs, or by modifying the embedded DPLL solver so that to apply basic propositional techniques to produce an unsat core. In particular, we are not aware of any work aiming at producing small unsatisfiable cores in SMT. In this paper we present a novel approach addressing this problem. The main idea is to combine an SMT solver with an external propositional core extractor. The SMT solver stores and returns the theory lemmas it had to prove in order to refute the input formula; the external core extractor is then called on the boolean abstraction of the original SMT problem and of the theory lemmas. The resulting boolean unsatisfiable core is cleaned from (the boolean abstraction of) all theory lemmas, and it is refined back into a subset of the original clauses. The result is an unsatisfiable core of the original SMT problem. Although simple in principle, the approach is conceptually interesting: basically, the SMT solver is used to dynamically lift the suitable amount of theory information to the boolean level. Furthermore, the approach has several advantages in practice: first, it is extremely simple to implement and to update; second, it is effective in finding small cores; third, the core extraction is not prone to complex SMT reasoning; finally, it can be interfaced with every propositional core extractor in a plug-and-play manner, so that to benefit for free of all unsat-core reduction techniques which have been or will be made available. For lack of space, in this short version of the paper we omit many details, any related work and the description and the results of our extensive experimental evaluation of the approach. They can be found in the extended version of the paper [3].