## Stochastic Boolean Satisfiability (2000)

Venue: | Journal of Automated Reasoning |

Citations: | 50 - 5 self |

### BibTeX

@ARTICLE{Littman00stochasticboolean,

author = {Michael L. Littman and Stephen M. Majercik and Toniann Pitassi},

title = {Stochastic Boolean Satisfiability},

journal = {Journal of Automated Reasoning},

year = {2000},

volume = {27},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSat, which can function for probabilistic domains as Sat does for deterministic domains. It shows the connection between SSat and well studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSat instances. These algorithms are validated on random SSat formulae generated under the fixed-clause model. In spite of the large complexity gap between SSat (PSPACE) and Sat (NP), the paper suggests that much of what we've learned about Sat transfers to the probabilistic domain. 1. Introduction There has been a recent focus in artificial intelligence (AI) on solving problems exh...

### Citations

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Citation Context ...answers to SSat problems (existential and randomized quantifiers only). 2.1. A Davis-Putnam-Logemann-Loveland Algorithm The Davis-Putnam-Logemann-Loveland (DPLL) algorithm for Boolean satisfiability (=-=Davis, Logemann, & Loveland, 1962-=-) works by enumerating partial assignments and monitoring for opportunities to simplify the formula. These simplifications, or pruning rules, make it possible to solve problems whose entire set of ass... |

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Citation Context ...Sat problems. In part due to the success of solving satisfiability problems, researchers are finding efficient ways of modeling diverse problems within the satisfiability framework, such as planning (=-=Kautz & Selman, 1996-=-). A formal definition of the Sat decision problem follows. Let x = hx 1 ; x 2 ; : : : ; x n i be a collection of n Boolean (0/1) variables, and OE(x) be a k-CNF Boolean formula on these variables wit... |

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Citation Context ...mma1 ; R xn(E[OE(x)]s`) influence diagrams best polynomial-horizon plan planning problems has been exploited to create a competitive planning algorithm (Majercik & Littman, 1999). Influence diagrams (=-=Shachter, 1986-=-) are a belief-network-like representation for the same type of problem. Table II summarizes the relations between the complexity classes, stochastic satisfiability problems, belief network problems, ... |

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Citation Context ... that some of the variables are under the control of "nature." This is closely related to solving a finite-horizon partially observable Markov decision process (pomdp), which is also PSPACE-=-=complete (Papadimitriou & Tsitsiklis, 1987). Th-=-e problem remains PSPACE-complete when the domain is specified compactly via probabilistic strips operators or an equivalent representation (Mundhenk et al., 1997), even if the domain is "fully o... |

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Citation Context ...PACE. The NP-complete problem Sat is the SSat problem obtained by using only existential quantifiers and setting ` = 1. The problem of finding the most probable explanation (MPE) in a belief network (=-=Dechter, 1996-=-) is (polynomially) equivalent to Sat. A related problem from planning under uncertainty is determining whether there is some choice of actions such that the most likely trajectory through state space... |

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Citation Context ...y 21 when a satisfying assignment does not exist. But, compared to systematic solvers, randomized local search algorithms can solve random satisfiability problems that are orders of magnitude larger (=-=Selman, Kautz, & Cohen, 1996-=-). For Majsat problems, random sampling is an obvious approach. Some number of assignments to the random variables are generated according to their probabilities; the probability that the formula is s... |

219 | On the hardness of approximate reasoning
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Citation Context ...e for the well-known complexity class NP, and are therefore formally equivalent to Sat, many planning and reasoning problems in probabilistic models lie in other complexity classes, such as #P or PP (=-=Roth, 1996-=-; Littman, Goldsmith, & Mundhenk, 1998). This means that, with respect to the current state of complexity theory, these problems cannot be polynomially reduced to Sat. However, many standard uncertain... |

205 | Using CSP look-back techniques to solve real-world SAT instances
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Citation Context ...reme case; other ideas are needed to address this more generally. Considerations such as this have prompted a great deal of research into efficient splitting heuristics for Sat (Hooker & Vinay, 1994; =-=Bayardo & Schrag, 1997-=-; Gallo & Urbani, 1989; Harche, Hooker, & Thompson, 1994; Jeroslow & Wang, 1990; Li & Anbulagan, 1997; Crawford & Auton, 1996; Freeman, 1995). An appropriate splitting heuristic can reduce the running... |

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Citation Context ... k = 3 occurs at around 4:2, and a recent theoretical result by Friedgut (1997) shows that for each n, there is a sharp 0--1 threshold ff k (n) that possibly varies with n. It is known that ff 2 = 1 (=-=Chv'atal & Reed, 1992-=-; Goerdt, 1996), and that for k = 3, 3:003sff k (n)s4:601 (Frieze & Suen, 1996; Kirousis et al., 1998). An interesting feature of the threshold is that the dual problems of bounding the threshold valu... |

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Citation Context ...plitting heuristics for Sat (Hooker & Vinay, 1994; Bayardo & Schrag, 1997; Gallo & Urbani, 1989; Harche, Hooker, & Thompson, 1994; Jeroslow & Wang, 1990; Li & Anbulagan, 1997; Crawford & Auton, 1996; =-=Freeman, 1995-=-). An appropriate splitting heuristic can reduce the running time of the DPLL algorithm by several orders of magnitude. But, do these splitting heuristics improve efficiency in SSat problems? Relative... |

153 | The complexity of stochastic games - Condon - 1992 |

139 | An Algorithm to Evaluate Quantified Boolean Formulae - Cadoli, Giovanardi, et al. - 1998 |

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Citation Context ...esearch into efficient splitting heuristics for Sat (Hooker & Vinay, 1994; Bayardo & Schrag, 1997; Gallo & Urbani, 1989; Harche, Hooker, & Thompson, 1994; Jeroslow & Wang, 1990; Li & Anbulagan, 1997; =-=Crawford & Auton, 1996-=-; Freeman, 1995). An appropriate splitting heuristic can reduce the running time of the DPLL algorithm by several orders of magnitude. But, do these splitting heuristics improve efficiency in SSat pro... |

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Citation Context ...shows that for each n, there is a sharp 0--1 threshold ff k (n) that possibly varies with n. It is known that ff 2 = 1 (Chv'atal & Reed, 1992; Goerdt, 1996), and that for k = 3, 3:003sff k (n)s4:601 (=-=Frieze & Suen, 1996-=-; Kirousis et al., 1998). An interesting feature of the threshold is that the dual problems of bounding the threshold value from above and below appear to be asymmetric. At slightly below the 0--1 thr... |

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Citation Context ...of randomly generated formulae. Second, using an appropriate choice of parameters, randomly chosen formulae are empirically difficult for satisfiability algorithms, and are a commonly used benchmark (=-=Gu et al., 1997-=-). The next section briefly reviews some analytical results concerning random Sat and Majsat instances. 1.4.1. Random Sat An interesting property of random k-CNF formulae is that when m is small relat... |

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Citation Context ...pted a great deal of research into efficient splitting heuristics for Sat (Hooker & Vinay, 1994; Bayardo & Schrag, 1997; Gallo & Urbani, 1989; Harche, Hooker, & Thompson, 1994; Jeroslow & Wang, 1990; =-=Li & Anbulagan, 1997-=-; Crawford & Auton, 1996; Freeman, 1995). An appropriate splitting heuristic can reduce the running time of the DPLL algorithm by several orders of magnitude. But, do these splitting heuristics improv... |

113 |
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Citation Context ... such as this have prompted a great deal of research into efficient splitting heuristics for Sat (Hooker & Vinay, 1994; Bayardo & Schrag, 1997; Gallo & Urbani, 1989; Harche, Hooker, & Thompson, 1994; =-=Jeroslow & Wang, 1990-=-; Li & Anbulagan, 1997; Crawford & Auton, 1996; Freeman, 1995). An appropriate splitting heuristic can reduce the running time of the DPLL algorithm by several orders of magnitude. But, do these split... |

103 |
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Citation Context ... 4:2, and a recent theoretical result by Friedgut (1997) shows that for each n, there is a sharp 0--1 threshold ff k (n) that possibly varies with n. It is known that ff 2 = 1 (Chv'atal & Reed, 1992; =-=Goerdt, 1996-=-), and that for k = 3, 3:003sff k (n)s4:601 (Frieze & Suen, 1996; Kirousis et al., 1998). An interesting feature of the threshold is that the dual problems of bounding the threshold value from above a... |

102 | Games against nature - Papadimitriou - 1985 |

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Citation Context ...method (including the DPLL procedure that forms the basis of the algorithms in this paper) will succeed in sub-exponential time on a broad range of randomly generated formulae (Chv'atal & Reed, 1992; =-=Beame & Pitassi, 1996-=-; Beame et al., 1998). In particular, Beame et al. (1998) show that for k = 3 and msn 2 = log n, almost certainly a random formula has a polynomial-size decision tree, but for m ! n 6=5 , almost certa... |

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Citation Context ...h that a random k-CNF formula from F k;n m is almost certainly satisfiable for m=n ! ff k (as n gets large), and almost certainly unsatisfiable if m=n ? ff k . As mentioned above, empirical evidence (=-=Selman, Mitchell, & Levesque, 1996-=-) suggests that this 0--1 threshold value for k = 3 occurs at around 4:2, and a recent theoretical result by Friedgut (1997) shows that for each n, there is a sharp 0--1 threshold ff k (n) that possib... |

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Citation Context ...ection 3.1 confirm the form of Equation 1. While the above calculation gives an upper bound on m as a function of t and n, this upper bound is likely larger than the actual number. In several papers (=-=Kamath et al., 1995-=-; Kirousis et al., 1998), it has been demonstrated that the expected number of satisfying assignments is larger than the number of satisfying assignments almost certainly, because for those formulae t... |

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Citation Context ..., there is a sharp 0--1 threshold ff k (n) that possibly varies with n. It is known that ff 2 = 1 (Chv'atal & Reed, 1992; Goerdt, 1996), and that for k = 3, 3:003sff k (n)s4:601 (Frieze & Suen, 1996; =-=Kirousis et al., 1998-=-). An interesting feature of the threshold is that the dual problems of bounding the threshold value from above and below appear to be asymmetric. At slightly below the 0--1 threshold, there are simpl... |

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Citation Context ...em remains PSPACE-complete when the domain is specified compactly via probabilistic strips operators or an equivalent representation (Mundhenk et al., 1997), even if the domain is "fully observab=-=le" (Littman, 1997-=-). The connection between SSat and these planning problems has been exploited to create a competitive planpaper. tex; 2/08/1999; 17:08; p.7 8 Littman, Majercik, and Pitassi Table II. Different arrange... |

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Citation Context ....1.1, handles this extreme case; other ideas are needed to address this more generally. Considerations such as this have prompted a great deal of research into efficient splitting heuristics for Sat (=-=Hooker & Vinay, 1994-=-; Bayardo & Schrag, 1997; Gallo & Urbani, 1989; Harche, Hooker, & Thompson, 1994; Jeroslow & Wang, 1990; Li & Anbulagan, 1997; Crawford & Auton, 1996; Freeman, 1995). An appropriate splitting heuristi... |

70 | MAXPLAN: A new approach to probabilistic planning
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Citation Context ...stic problems using techniques for testing Boolean satisfiability. Some recent work has looked at combinations of these ideas, viewing planning under uncertainty as stochastic Boolean satisfiability (=-=Majercik & Littman, 1998-=-). This paper extends these ideas, and provides a general approach for combining reasoning about uncertainty and satisfiability. The remainder of this section reviews deterministic satisfiability, int... |

64 |
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Citation Context ...re needed to address this more generally. Considerations such as this have prompted a great deal of research into efficient splitting heuristics for Sat (Hooker & Vinay, 1994; Bayardo & Schrag, 1997; =-=Gallo & Urbani, 1989-=-; Harche, Hooker, & Thompson, 1994; Jeroslow & Wang, 1990; Li & Anbulagan, 1997; Crawford & Auton, 1996; Freeman, 1995). An appropriate splitting heuristic can reduce the running time of the DPLL algo... |

62 | Satisfiability coding lemma
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Citation Context ...ecent flurry of activity in analyzing common Davis-Putnam algorithms, and proving that these algorithms are guaranteed to solve any k-SAT instance in subexponential-time (Monien & Speckenmeyer, 1985; =-=Paturi, Pudlak, & Zane, 1997-=-; Paturi et al., 1998). (For example, a very simple randomized DPLL-type algorithm solves 3-Sat in worst-case time 2 (2=3)n .) It would be interesting to show that these analyses carry over to the SSa... |

59 | M.L.: Contingent planning under uncertainty via stochastic satisfiability
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Citation Context ...`) maximum a posteriori hypothesis best polynomial-size plan PSPACE Alternating SSat 9x1 ; R x2 ; : : : ; 9xn\Gamma1 ; R xn(E[OE(x)]s`) influence diagrams best polynomial-horizon plan ning algorithm (=-=Majercik & Littman, 1999-=-). Influence diagrams are a belief-network-like representation for the same type of problem. Table II summarizes the relations between the complexity classes, stochastic satisfiability problems, belie... |

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44 | Beyond NP: The QSAT Phase Transition - Gent, Walsh - 1999 |

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Citation Context ...his more generally. Considerations such as this have prompted a great deal of research into efficient splitting heuristics for Sat (Hooker & Vinay, 1994; Bayardo & Schrag, 1997; Gallo & Urbani, 1989; =-=Harche, Hooker, & Thompson, 1994-=-; Jeroslow & Wang, 1990; Li & Anbulagan, 1997; Crawford & Auton, 1996; Freeman, 1995). An appropriate splitting heuristic can reduce the running time of the DPLL algorithm by several orders of magnitu... |

23 | Threshold phenomena in random graph colouring and satisfiability - Achlioptas - 1999 |

22 |
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Citation Context ... methods for solving satisfiability instances. In particular, algebraic approaches based on discrete versions of the Grobner basis algorithm are looking promising for solving satisfiability problems (=-=Clegg, Edmonds, & Impagliazzo, 1996-=-). Future work will examine to what extent these methods carry over to the SSat domain. A number of recent studies have shown that the satisfiability framework is useful for examining a variety of sea... |

18 | Experimental analysis of the computational cost of evaluating quantified Boolean formulae - Cadoli, Giovanardi, et al. - 1997 |

15 |
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Citation Context ...ll-known complexity class NP, and are therefore formally equivalent to Sat, many planning and reasoning problems in probabilistic models lie in other complexity classes, such as #P or PP (Roth, 1996; =-=Littman, Goldsmith, & Mundhenk, 1998-=-). This means that, with respect to the current state of complexity theory, these problems cannot be polynopaper. tex; 1/09/1999; 0:14; p.5 6 Littman, Majercik, and Pitassi mially reduced to Sat. Howe... |

14 |
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Citation Context ...algorithm. There has been a recent flurry of activity in analyzing common Davis-Putnam algorithms, and proving that these algorithms are guaranteed to solve any k-SAT instance in subexponential-time (=-=Monien & Speckenmeyer, 1985-=-; Paturi, Pudlak, & Zane, 1997; Paturi et al., 1998). (For example, a very simple randomized DPLL-type algorithm solves 3-Sat in worst-case time 2 (2=3)n .) It would be interesting to show that these ... |

13 |
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Citation Context ...also PSPACE-complete (Papadimitriou & Tsitsiklis, 1987). The problem remains PSPACE-complete when the domain is specified compactly via probabilistic strips operators or an equivalent representation (=-=Mundhenk et al., 1997), even if-=- the domain is "fully observable" (Littman, 1997). The connection between SSat and these planning problems has been exploited to create a competitive planpaper. tex; 2/08/1999; 17:08; p.7 8 ... |

10 |
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Citation Context ...PLL procedure that forms the basis of the algorithms in this paper) will succeed in sub-exponential time on a broad range of randomly generated formulae (Chv'atal & Reed, 1992; Beame & Pitassi, 1996; =-=Beame et al., 1998-=-). In particular, Beame et al. (1998) show that for k = 3 and msn 2 = log n, almost certainly a random formula has a polynomial-size decision tree, but for m ! n 6=5 , almost certainly a random formul... |

8 | Nonapproximability results for Markov decision processes - Lusena, Goldsmith, et al. - 1998 |

2 |
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Citation Context ...alyzing common Davis-Putnam algorithms, and proving that these algorithms are guaranteed to solve any k-SAT instance in subexponential-time (Monien & Speckenmeyer, 1985; Paturi, Pudlak, & Zane, 1997; =-=Paturi et al., 1998-=-). (For example, a very simple randomized DPLL-type algorithm solves 3-Sat in worst-case time 2 (2=3)n .) It would be interesting to show that these analyses carry over to the SSat setting. In additio... |