Solving Exist-Random Quantified Stochastic Boolean Satisfiability via Clause Selection

Abstract

Stochastic Boolean satisfiability (SSAT) is an expressive language to formulate decision problems with randomness. Solving SSAT formulas has the same PSPACE-complete computational complexity as solving quantified Boolean formulas (QBFs). Despite its broad applications and profound theoretical values, SSAT has received relatively little attention compared to QBF. In this paper, we focus on exist-random quantified SSAT formulas, also known as E-MAJSAT, which is a special fragment of SSAT commonly applied in probabilistic conformant planning, posteriori hypothesis, and maximum expected utility. Based on clause selection, a recently proposed QBF technique, we propose an algorithm to solve E-MAJSAT. Moreover, our method can provide an approximate solution to E-MAJSAT with a lower bound when an exact answer is too expensive to compute. Experiments show that the proposed algorithm achieves significant performance gains and memory savings over the state-of-the-art SSAT solvers on a number of benchmark formulas, and provides useful lower bounds for cases where prior methods fail to compute exact answers.

DPO: Dynamic-Programming Optimization on Hybrid Constraints

“…For future work, we plan to add support for hybrid inputs, such as PB and cardinality constraints. Also, DPO can be extended to solve more general problems, e.g., existential-random stochastic satisfiability [Lee et al, 2018], maximum model counting [Fremont et al, 2017], and functional aggregate queries [Abo Khamis et al, 2016]. Another research direction is to improve DPO with parallelism, as a portfolio solver (e.g., [Xu et al, 2008]) or with a multi-core ADD package (e.g., [van Dijk and van de Pol, 2015]).…”

DPER: Dynamic Programming for Exist-Random Stochastic SAT

“…The ER-SSAT solver erSSAT [Lee et al, 2018] uses clause-containment learning, a technique inspired by clause selection [Janota and Marques-Silva, 2015] in QBF evaluation. Clause-containment learning prunes the search space with blocking clauses that are deduced after trying some truth assignments.…”

“…Also having that form is the exist-random stochastic satisfiability (ER-SSAT) problem. Given a Boolean formula ϕ over X ∪ Y , ER-SSAT requests a truth assignment τ X for X that maximizes the sum of weights of models τ Y of the residual formula ϕ τ X over Y [Lee et al, 2018]. ER-SSAT can be viewed as a generalization of SAT and WMC.…”

“…By building on techniques from WPMC [Dudek et al, 2021], Boolean MPE [Phan and Vardi, 2022], and MaxSAT [Kyrillidis et al, 2022], we construct DPER, a dynamic-programming ER-SSAT framework. We then compare DPER to state-of-the-art solvers (DC-SSAT [Majercik and Boots, 2005] and erSSAT [Lee et al, 2018]). We observe that DPER contributes to the portfolio of exact ER-SSAT solvers, through very competitive performance on instances with low-width project-join trees.…”

CAQE and QuAbS: Abstraction Based QBF Solvers

“…Clausal Abstraction is a solving method for quantified Boolean formulas that was independently developed by Janota & Marques-Silva [46] 1. and Rabe & Tentrup [66]. While initially only applicable to QBFs in prenex conjunctive normal form, there have been extensions to QBFs in negation normal form [36], parallelization [71], satisfiability modulo theories [13], quantified stochastic Boolean satisfiability [51], and dependency quantified Boolean formulas [73]. The underlying idea of clausal abstraction is to assign variables, where the assignment order is determined by the quantifier prefix, until either all clauses are satisfied or there is a set of clauses that cannot be satisfied at the same time.…”