Efficient theory combination via boolean search

Many approaches to deciding the satisfiability of quantifier-free formulae with respect to a background theory T—also known as SatisfiabilityModulo Theory, or SMT(T)—rely on the integration between an enumerator of truth assignments and a decision procedure for conjunction of literals in T. When the background theory T is the combination T1 ∪ T2 of two simpler theories, the approach is typically instantiated by means of a theory combination schema (e.g. Nelson–Oppen, Shostak). In this paper we propose a new approach to SMT(T1 ∪ T2), where the enumerator of truth assignments is integrated with two decision procedures, one for T1 and one for T2, acting independently from each other. The key idea is to search for a truth assignment not only to the atoms occurring in the formula, but also to all the equalities between variables which are shared between the theories. This approach is simple and expressive: for instance, no modification is required to handle non-convex theories (as opposed to traditional Nelson–Oppen combinations which require a mechanism for splitting). Furthermore, it can be made practical by leveraging on state-of-the-art boolean and SMT This work has been partly supported by ISAAC, an European sponsored project, contract no. AST3-CT-2003-501848, by ORCHID, a project sponsored by Provincia Autonoma di Trento, and by a grant from Intel Corporation. The work of T. Junttila has also been supported by the Academy of Finland, projects 53695 and 211025. ∗ Corresponding author. Fax: +39 0461 314 591. E-mail addresses: [email protected] (M. Bozzano), [email protected] (R. Bruttomesso), [email protected] (A. Cimatti), [email protected] (T. Junttila), [email protected] (S. Ranise), [email protected] (P. van Rossum), roberto. [email protected] (R. Sebastiani). 0890-5401/$ see front matter © 2006 Published by Elsevier Inc. doi:10.1016/j.ic.2005.05.011 1494 M. Bozzano et al. / Information and Computation 204 (2006) 1493–1525 search techniques, and on theory layering (i.e., cheaper reasoning first, andmore often). We provide thorough experimental evidence to support our claims: we instantiate the framework with two decision procedures for the combinations of Equality and Uninterpreted Functions (EUF) and Linear Arithmetic (LA), both for (the convex case of) reals and for (the non-convex case of) integers; we analyze the impact of the different optimizations on a variety of test cases; and we compare the approach with state-of-the-art competitor tools, showing that our implemented tool compares positively with them, sometimes with dramatic gains in performance. © 2006 Published by Elsevier Inc.