GANAK: A Scalable Probabilistic Exact Model Counter

Abstract

Given a Boolean formula F, the problem of model counting, also referred to as #SAT, seeks to compute the number of solutions of F. Model counting is a fundamental problem with a wide variety of applications ranging from planning, quantified information flow to probabilistic reasoning and the like. The modern #SAT solvers tend to be either based on static decomposition, dynamic decomposition, or a hybrid of the two. Despite dynamic decomposition based #SAT solvers sharing much of their architecture with SAT solvers, the core design and heuristics of dynamic decomposition-based #SAT solvers has remained constant for over a decade. In this paper, we revisit the architecture of the state-of-the-art dynamic decomposition-based #SAT tool, sharpSAT, and demonstrate that by introducing a new notion of probabilistic component caching and the usage of universal hashing for exact model counting along with the development of several new heuristics can lead to significant performance improvement over state-of-the-art model-counters. In particular, we develop GANAK, a new scalable probabilistic exact model counter that outperforms state-of-the-art exact and approximate model counters sharpSAT and ApproxMC3 respectively, both in terms of PAR-2 score and the number of instances solved. Furthermore, in our experiments, the model count returned by GANAK was equal to the exact model count for all the benchmarks. Finally, we observe that recently proposed preprocessing techniques for model counting benefit exact model counters while hurting the performance of approximate model counters.

Exploiting Database Management Systems and Treewidth for Counting

“…Further, the exploration of bitmap indices, as available in Oracle enterprise DBMS would be worth trying in our case (and for #Sat), since one can efficiently combine database columns by using extremely efficient bit operations. It might be worth to rigorously test and explore our extensions on limiting the number of rows per table for approximating #Sat or other counting problems, cf., [8,19,39]. Another interesting research direction is to study whether efficient data representation techniques on DBMS can be combined with dynamic programming in order to lift our solver to quantified Boolean formulas.…”

A Study of Symmetry Breaking Predicates and Model Counting

“…Consider specifying the well-known n-Queens problem of placing n interchangeable queens 4 on a fixed n×n chess-board, and computing the number of solutions to the problem using a modern propositional model counter [16,40,50]. Figure 1 shows a fragment of an Alloy specification of the n-Queens problem, which has been studied before using Alloy [2,4,55].…”

“…Recently, Lagniez and Marquis proposed a recursive algorithm, called ProjMC [40], that exploits the disjunctive decomposition technique pioneered in earlier works to perform projected model counting. Concurrently, another approach, called Ganak [50], for projected model counting has been developed that provides probabilistic exact bounds via usage of universal hash functions. In this work, we focus on ProjMC due to its ability to provide exact counts and demonstrated scalability in comparison to other approaches.…”

“…Propositional model counting is the classic problem of counting the number of all solutions for the given formula in propositional logic. While the core problem is an integral part of complexity theory literature, advances in propositional satisfiability (SAT) solvers and other decision procedures in the last decade have led to much progress in tackling this problem in innovative ways [7,9,10,15,17,31,39,40,47,49,50,56,64]. These advances have fueled the application of model counters in various software verification and reliability domains, e.g., to perform probabilistic analyses [13,26,28], check and repair string manipulation code [9,41], and estimate information leakage using quantified information flow [19,44].…”

“…As a result, the translated formula is equisatisfiable to the original formula but may not be equivalent to it, and hence it may be the case that the model count for the CNF formula is very different from the original formula. Several modern model counters [16,40,50] readily handle this case by providing support for projected model counting [10], i.e., computing the model count with respect to a subset of all the variables. For Alloy, the subset is the primary variables, i.e., all boolean variables that directly correspond to the variables in the Alloy specification.…”