Estimating the Volume of the Solution Space of SMT(LIA) Constraints by a Flat Histogram Method

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.

Keywords

• Markov Chain Monte-Carlo
• SAT modulo theories
• volume computation
• flat histogram