Lazy Satisfiability Modulo Theories

Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a first-order formula with respect to some decidable first-order theory T(SMT(T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingly important due to its applications in many domains in different communities, in particular in formal verification. An amount of papers with novel and very efficient techniques for SMT has been published in the last years, and some very efficient SMT tools are now available. Typical SMT(T) problems require testing the satisfiability of formulas which are boolean combinations of atomic propositions and atomic expressions in T, so that heavy boolean reasoning must be efficiently combined with expressive theory-specific reasoning. The dominating approach to SMT(T), called lazy approach, is based on the integration of a SAT solver and of a decision procedure able to handle sets of atomic constraints in T (T-solver), handling respectively the boolean and the theory-specific components of reasoning. Unfortunately, neither the problem of building an efficient SMT solver, nor even that of acquiring a comprehensive background knowledge in lazy SMT, is of simple solution. In this paper we present an extensive survey of SMT, with particular focus on the lazy approach. We survey, classify and analyze from a theory-independent perspective the most effective techniques and optimizations which are of interest for lazy SMT and which have been proposed in various communities; we discuss their relative benefits and drawbacks; we provide some guidelines about their choice and usage; we also analyze the features for SAT solvers and T-solvers which make them more suitable for an integration. The ultimate goals of this paper are to become a source of a common background knowledge and terminology for students and researchers in different areas, to provide a domain-independent reference guide for developers of SMT tools, and to stimulate the cross-fertilization of techniques and ideas among different communities.