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.
Stochastic Boolean Satisfiability (SSAT) is a powerful formalism to represent computational problems with uncertainty, such as belief network inference and propositional probabilistic planning. Solving SSAT formulas lies in the PSPACEcomplete complexity class same as solving Quantified Boolean Formulas (QBFs). While many endeavors have been made to enhance QBF solving in recent years, SSAT has drawn relatively less attention. This paper focuses on random-exist quantified SSAT formulas, and proposes an algorithm combining modern satisfiability (SAT) techniques and model counting to improve computational efficiency. Unlike prior exact SSAT algorithms, the proposed method can be easily modified to solve approximate SSAT by deriving upper and lower bounds of satisfying probability. Experimental results show that our method outperforms the stateof-the-art algorithm on random k-CNF and AIrelated formulas in both runtime and memory usage, and has effective application to approximate SSAT on VLSI circuit benchmarks.
Stochastic Boolean Satisfiability (SSAT) is a logical formalism to model decision problems with uncertainty, such as Partially Observable Markov Decision Process (POMDP) for verification of probabilistic systems. SSAT, however, is limited by its descriptive power within the PSPACE complexity class. More complex problems, such as the NEXPTIME-complete Decentralized POMDP (Dec-POMDP), cannot be succinctly encoded with SSAT. To provide a logical formalism of such problems, we generalize the Dependency Quantified Boolean Formula (DQBF), a representative problem in the NEXPTIME-complete class, to its stochastic variant, named Dependency SSAT (DSSAT), and show that DSSAT is also NEXPTIME-complete. We demonstrate the potential applications of DSSAT to circuit synthesis of probabilistic and approximate design. Furthermore, to study the descriptive power of DSSAT, we establish a polynomial-time reduction from Dec-POMDP to DSSAT. With the theoretical foundations paved in this work, we hope to encourage the development of DSSAT solvers for potential broad applications.
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