Computing and estimating the volume of the solution space of SMT(LA) constraints

The satisfiability modulo theories (SMT) problem is a decision problem, i.e., deciding the satisfiability of logical formulas with respect to combinations of background theories (like reals, integers, arrays, bit-vectors). In this paper, we study the counting version of SMT with respect to linear arithmetic - SMT(LA), which generalizes both model counting and volume computation of convex polytopes. We describe a method for estimating the volume of convex polytopes based on the Multiphase Monte-Carlo method. It employs a new technique to reutilize random points, so that the number of random points can be significantly reduced. We prove that the reutilization technique has no side-effect on the error. We also investigate a simplified version of hit-and-run method: the coordinate directions method. Based on volume estimation method for polytopes, we present an approach to estimating the volume of the solution space of SMT(LA) formulas. It employs a heuristic strategy to accelerate the volume estimation procedure. In addition, we devise some specific techniques for instances that arise from program analysis.