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

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.

[1]  D. Landau,et al.  Efficient, multiple-range random walk algorithm to calculate the density of states. , 2000, Physical review letters.

[2]  Jun Yan,et al.  An efficient method to generate feasible paths for basis path testing , 2008, Inf. Process. Lett..

[3]  Martin E. Dyer,et al.  On the Complexity of Computing the Volume of a Polyhedron , 1988, SIAM J. Comput..

[4]  Joost-Pieter Katoen,et al.  A compositional modelling and analysis framework for stochastic hybrid systems , 2012, Formal Methods in System Design.

[5]  Min Zhou,et al.  Estimating the Volume of Solution Space for Satisfiability Modulo Linear Real Arithmetic , 2014, Theory of Computing Systems.

[6]  Peng Zhang,et al.  Computing and estimating the volume of the solution space of SMT(LA) constraints , 2018, Theor. Comput. Sci..

[7]  Sharad Malik,et al.  Performance analysis of embedded software using implicit path enumeration , 1997, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[8]  Jesús A. De Loera,et al.  Effective lattice point counting in rational convex polytopes , 2004, J. Symb. Comput..

[9]  R. Belardinelli,et al.  Fast algorithm to calculate density of states. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Pierre E. Jacob,et al.  The Wang-Landau algorithm reaches the Flat Histogram criterion in finite time , 2014 .

[11]  Juan J de Pablo,et al.  Sculpting bespoke mountains: Determining free energies with basis expansions. , 2015, The Journal of chemical physics.

[12]  V. A. Ivanov,et al.  Multidimensional stochastic approximation Monte Carlo. , 2016, Physical review. E.

[13]  Juan J de Pablo,et al.  Basis function sampling: a new paradigm for material property computation. , 2014, Physical review letters.

[14]  Myra B. Cohen,et al.  An orchestrated survey of methodologies for automated software test case generation , 2013, J. Syst. Softw..

[15]  Jian Zhang,et al.  A Constraint Solver and Its Application to Path Feasibility Analysis , 2001, Int. J. Softw. Eng. Knowl. Eng..