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Open AccessArticle

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

by 1, 2, 3 and 4,*
1
School of Astronautics, Beihang University, Beijing 100191, China
2
Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China
3
Beijing Aerospace Control Center, Beijing 100094, China
4
Hubei Provincial Key Laboratory of Intelligent Robot, Wuhan Institute of Technology, Wuhan 430205, China
*
Author to whom correspondence should be addressed.
Algorithms 2018, 11(9), 142; https://doi.org/10.3390/a11090142
Received: 12 June 2018 / Revised: 12 September 2018 / Accepted: 14 September 2018 / Published: 18 September 2018
(This article belongs to the Special Issue Parameter Estimation Algorithms and Its Applications)
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. View Full-Text
Keywords: Markov Chain Monte-Carlo; SAT modulo theories; volume computation; flat histogram Markov Chain Monte-Carlo; SAT modulo theories; volume computation; flat histogram
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MDPI and ACS Style

Gao, W.; Lv, H.; Zhang, Q.; Cai, D. Estimating the Volume of the Solution Space of SMT(LIA) Constraints by a Flat Histogram Method. Algorithms 2018, 11, 142. https://doi.org/10.3390/a11090142

AMA Style

Gao W, Lv H, Zhang Q, Cai D. Estimating the Volume of the Solution Space of SMT(LIA) Constraints by a Flat Histogram Method. Algorithms. 2018; 11(9):142. https://doi.org/10.3390/a11090142

Chicago/Turabian Style

Gao, Wei; Lv, Hengyi; Zhang, Qiang; Cai, Dunbo. 2018. "Estimating the Volume of the Solution Space of SMT(LIA) Constraints by a Flat Histogram Method" Algorithms 11, no. 9: 142. https://doi.org/10.3390/a11090142

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