Abstract
The satisfiability modulo theories (SMT) problem is to decide the satisfiability of a logical formula with respect to a given background theory. This work studies the counting version of SMT with respect to linear integer arithmetic (LIA), termed SMT(LIA). Specifically, the purpose of this paper is to count the number of solutions (volume) of a SMT(LIA) formula, which has many important applications and is computationally hard. To solve the counting problem, an approximate method that employs a recent Markov Chain Monte Carlo (MCMC) sampling strategy called “flat histogram” is proposed. Furthermore, two refinement strategies are proposed for the sampling process and result in two algorithms, MCMC-Flat1/2 and MCMC-Flat1/t, respectively. In MCMC-Flat1/t, a pseudo sampling strategy is introduced to evaluate the flatness of histograms. Experimental results show that our MCMC-Flat1/t method can achieve good accuracy on both structured and random instances, and our MCMC-Flat1/2 is scalable for instances of convex bodies with up to 7 variables.
Highlights
Satisfiability Modulo Theories (SMT) considers the satisfiability of a formula specified in a fragment of the first order logic with some kind of background theory [1]
To estimate the volume of an SMT(LIA) formula, we propose a flat histogram method that consists of three components
We propose a Markov Chain Monte Carlo (MCMC) method to solve the #SMT(LIA) problem based on the flat histogram method
Summary
Satisfiability Modulo Theories (SMT) considers the satisfiability of a formula specified in a fragment of the first order logic with some kind of background theory [1]. One of those important theories is linear integer arithmetic (LIA). An atom in a SMT(LIA) formula is an inequation of LIA. Counting the number of models of a SMT(LIA) formula, #SMT(LIA), is theoretically hard. If all the LIA constraints are joined only by conjunctions, #SMT(LIA) reduces to the problem of computing the volume of convex bodies [2] which is shown to be #P-Complete [3]. The #SMT(LIA) problem considered in this paper supports formulas with arbitrary Boolean structures
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