Bruttomesso, Roberto and Cimatti, Alessandro and Franzen, Anders and Griggio, Alberto and Sebastiani, Roberto (2008) Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: a Comparative Analysis. UNSPECIFIED. (Submitted)
Most state-of-the-art approaches for Satisfiability Modulo Theory (SMT(T)) rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory T (T-solver). Often T is the combination (T1 U T2) of two (or more) simpler theories (SMT(T1 U T2)), s.t. the specific Ti-Solvers must be combined. Up to a few years ago, the standard approach to SMT(T1 U T2) was to integrate the SAT solver with one combined (T1 U T2)-solver, obtained from two distinct Ti-Solvers by means of evolutions of Nelson and Oppen's (NO) combination procedure, in which the Ti-Solvers deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent SMT(T1 U T2) procedure called Delayed Theory Combination (DTC), in which each Ti-Solver interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for SMT(T1 U T2). On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the search. On the other hand, we show that the extra amount of Boolean search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the Ti-Solvers, which relate the amount of extra Boolean search required to the SAT solver by DTC with the number of deductions and case-splits required to the Ti-Solvers by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the Ti-Solvers required by NO, DTC causes no extra Boolean search; (ii) using Ti-Solvers with limited or no deduction capabilities, the amount of extra Boolean search required can be reduced down to a negligible amount by controlling the quality of the T-conflict sets returned by the T-solvers.
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