A Progressive Simplifier for Satisfiability Modulo Theories

Abstract

In this paper we present a new progressive cooperating simplifier for deciding the satisfiability of a quantifier-free formula in the first-order theory of integers involving combinations of sublogics, referred to as Satisfiability Modulo Theories (SMT). Our approach, given an SMT problem, replaces each non-propositional theory atom with a Boolean indicator variable yielding a purely propositional formula to be decided by a SAT solver. Starting with the most abstract representation (the Boolean formula), the solver gradually integrates more complex theory solvers into the working decision procedure. Additionally, we propose a method to simplify “expensive” atoms into suitable conjunctions of “cheaper” theory atoms when conflicts occur. This process considerably increases the efficiency of the overall procedure by reducing the number of calls to the slower theory solvers. This is made possible by adopting our novel inter-logic implication framework, as proposed in this paper. We have implemented these methods in our Ario SMT solver by combining three different theory solvers within a DPLL-style SAT solver: a Unit-Two-Variable-Per-Inequality (UTVPI) solver, an integer linear programming (ILP) solver, and a solver for systems of equalities with uninterpreted functions. The efficiencies of our proposed algorithms are demonstrated and exhaustively investigated on a wide range of benchmarks in hardware and software verification domain. Empirical results are also presented showing the advantages/limitations of our methods over other modern techniques for solving these SMT problems.