The Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI-21) Dependency Stochastic Boolean Satisfiability: A Logical Formalism for NEXPTIME Decision Problems with Uncertainty Nian-Ze Lee,1 Jie-Hong R. Jiang1, 2 1 Graduate Institute of Electronics Engineering, National Taiwan University 2 Department of Electrical Engineering, National Taiwan University No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan fd04943019, [email protected] Abstract to their simplicity and generality, various satisfiability for- mulations are under active investigation. Stochastic Boolean Satisfiability (SSAT) is a logical formal- ism to model decision problems with uncertainty, such as Among the quantified decision procedures, QBF and Partially Observable Markov Decision Process (POMDP) for SSAT are closely related. While SSAT extends QBF to verification of probabilistic systems. SSAT, however, is lim- allow random quantifiers to model uncertainty, they are ited by its descriptive power within the PSPACE complex- both PSPACE-complete (Stockmeyer and Meyer 1973). A ity class. More complex problems, such as the NEXPTIME- number of SSAT solvers have been developed and applied complete Decentralized POMDP (Dec-POMDP), cannot be in probabilistic planning, formal verification of probabilis- succinctly encoded with SSAT. To provide a logical formal- tic design, partially observable Markov decision process ism of such problems, we extend the Dependency Quantified (POMDP), and analysis of software security. For example, Boolean Formula (DQBF), a representative problem in the solver MAXPLAN (Majercik and Littman 1998) encodes a NEXPTIME-complete class, to its stochastic variant, named conformant planning problem as an exist-random quanti- Dependency SSAT (DSSAT), and show that DSSAT is also ZANDER NEXPTIME-complete. We demonstrate the potential appli- fied SSAT formula; solver (Majercik and Littman cations of DSSAT to circuit synthesis of probabilistic and 2003) deals with partially observable probabilistic planning approximate design. Furthermore, to study the descriptive by formulating the problem as a general SSAT formula; power of DSSAT, we establish a polynomial-time reduction solver DC-SSAT (Majercik and Boots 2005) relies on a from Dec-POMDP to DSSAT. With the theoretical founda- divide-and-conquer approach to speedup the solving of a tions paved in this work, we hope to encourage the develop- general SSAT formula. Solvers ressat and erssat (Lee, ment of DSSAT solvers for potential broad applications. Wang, and Jiang 2017, 2018) are developed for random- exist and exist-random quantified SSAT formulas respec- 1 Introduction tively, and applied to the formal verification of probabilistic design (Lee and Jiang 2018). POMDP has also been stud- Satisfiability (SAT) solvers (Biere, Heule, and van Maaren ied under the formalism of SSAT (Majercik and Littman 2009) have been successfully applied to numerous research 2003; Salmon and Poupart 2019). Recently, bi-directional fields including artificial intelligence (Nilsson 2014; Russell polynomial-time reductions between SSAT and POMDP are and Norvig 2016), electronic design automation (Marques- established (Salmon and Poupart 2019). The quantitative in- Silva and Sakallah 2000; Wang, Chang, and Cheng 2009), formation flow analysis for software security is also inves- software verification (Berard´ et al. 2013; Jhala and Majum- tigated as an exist-random quantified SSAT formula (Fre- dar 2009), etc. The tremendous benefits have encouraged the mont, Rabe, and Seshia 2017). development of more advanced decision procedures for sat- In view of the close relation between QBF and SSAT, isfiability with respect to more complex logics beyond pure we raise the question what would be the formalism that propositional. For example, solvers of the satisfiability mod- extends DQBF to the stochastic domain. We formalize the ulo theories (SMT) (De Moura and Bjørner 2011; Barrett dependency SSAT (DSSAT) as the answer to the question. and Tinelli 2018) accommodate first order logic fragments; We prove that DSSAT has the same NEXPTIME-complete quantified Boolean formula (QBF) (Narizzano, Pulina, and complexity as DQBF (Peterson, Reif, and Azhar 2001), and Tacchella 2006; Buning¨ and Bubeck 2009) allows both exis- therefore it can succinctly encode decision problems with tential and universal quantifiers; stochastic Boolean satisfia- uncertainty in the NEXPTIME complexity class. bilty (SSAT) (Littman, Majercik, and Pitassi 2001; Majercik To highlight the benefits of DSSAT over DQBF, we note 2009) models uncertainty with random quantification; and that DSSAT intrinsically represents an optimization problem dependency QBF (DQBF) (Balabanov, Chiang, and Jiang (the answer is the maximum satisfying probability) while 2014; Scholl and Wimmer 2018) equips Henkin quantifiers DQBF is a decision problem (the answer is either true or to describe multi-player games with partial information. Due false). The optimization nature of DSSAT potentially allows Copyright © 2021, Association for the Advancement of Artificial broader applications of the formalism. Moreover, DSSAT is Intelligence (www.aaai.org). All rights reserved. often preferable to DQBF in expressing problems involving 3877 uncertainty and probabilities. As case studies, we investigate 2.1 Stochastic Boolean Satisfiability its applicability in probabilistic system design/verification SSAT was first proposed by Papadimitriou and described as and artificial intelligence. games against nature (Papadimitriou 1985). An SSAT for- In system design of the post Moore’s law era, the practice Φ V = fx ; : : : ; x g mula over a set 1 n of variables isR of the p of very large scale integration (VLSI) circuit design experi- form: Q1x1;:::;Qnxn.φ, where each Qi 2 f9; g and ences a paradigm shift in design principles to overcome the φ V 9 Boolean formula over is quantifier-free.R Symbol de- obstacle of physical scaling of computation capacity. Prob- notes an existential quantifier, and p denotes a randomized abilistic design (Chakrapani et al. 2008) and approximate quantifier, which requires the probability that the quantified design (Venkatesan et al. 2011) are two such examples of variable equals > to be p 2 [0; 1]. Given an SSAT formula emerging design methodologies. The former does not re- Φ, the quantification structure Q1x1;:::;Qnxn is called the quire logic gates to be error-free, but rather allowing them prefix, and the quantifier-free Boolean formula φ is called the to function with probabilistic errors. The latter does not re- matrix. quire the implementation circuit to behave exactly the same Let x be the outermost variable in the prefix of an SSAT as the specification, but rather allowing their deviation to formula Φ. The satisfying probability of Φ, denoted by some extent. These relaxations to design requirements pro- Pr[Φ], is defined recursively by the following four rules: vide freedom for circuit simplification and optimization. We show that DSSAT can be a useful tool for the analysis of a) Pr[>] = 1, probabilistic design and approximate design. b) Pr[?] = 0, The theory and applications of Markov decision pro- c) Pr[Φ] = maxfPr[Φj:x]; Pr[Φjx]g, if x is existentially cess and its variants are among the most important topics quantified, in the study of artificial intelligence. For example, the de- Pr[Φ] = (1 − p) Pr[Φj ] + p Pr[Φj ] x d) R :x x , if is randomly cision problem involving multiple agents with uncertainty p and partial information is often considered as a decentral- quantified by , ized POMDP (Dec-POMDP) (Oliehoek, Amato et al. 2016). where Φj:x and Φjx denote the SSAT formulas obtained by The independent actions and observations of the individual eliminating the outermost quantifier of x via substituting the agents make POMDP for single-agent systems not applica- value of x in the matrix with ? and >, respectively. ble and require the more complex Dec-POMDP. Essentially The decision version of SSAT is stated as follows. Given the complexity is lifted from the PSPACE-complete policy an SSAT formula Φ and a threshold θ 2 [0; 1], decide evaluation of finite-horizon POMDP to the NEXPTIME- whether Pr[Φ] ≥ θ. On the other hand, the optimization ver- complete Dec-POMDP. We show that Dec-POMDP is poly- sion asks to compute Pr[Φ]. The decision version of SSAT nomial time reducible to DSSAT. was shown to be PSPACE-complete (Papadimitriou 1985). To sum up, the main results of this work include: 2.2 Dependency Quantified Boolean Formula • formulating the DSSAT problem (Section 3), DQBF was formulated as multiple-person alternation (Pe- • proving its NEXPTIME-completeness (Section 4), and terson and Reif 1979). In contrast to the linearly ordered • showing its applications in: prefix used in QBF, i.e., an existentially quantified variable will depend on all of its preceding universally quantified – analyzing probabilistic/approximate design (Section 5) variables, the quantification structure in DQBF is extended – modeling Dec-POMDP (Section 6). with Henkin quantifiers, where the dependency of an ex- Our results may encourage the development of DSSAT istentially quantified variable on the universally quantified solvers to enable potential broad applications. variables can be explicitly specified. A DQBF Φ over a set V = fx1; : : : ; xn; y1; : : : ; ymg of variables is of the form: 2 Preliminaries 8x ;:::; 8x ; 9y (D );:::; 9y (D ).φ, (1) In this section, we provide background knowledge about 1 n 1 y1 m ym SSAT, DQBF, probabilistic design, and Dec-POMDP. where each Dyj ⊆ fx1; : : : ; xng denotes the set of variables In the