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36
AND/OR Search Spaces for Graphical Models
, 2004
"... The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the gr ..."
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Cited by 102 (43 self)
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The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the graphical model explicitly and may sometime reduce the search space exponentially. Indeed, most
Value Elimination: Bayesian Inference via Backtracking Search
 IN UAI03
, 2003
"... We present Value Elimination, a new algorithm for Bayesian Inference. Given the same variable ordering information, Value Elimination can achieve performance that is within a constant factor of variable elimination or recursive conditioning, and on some problems it can perform exponentially bet ..."
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Cited by 49 (2 self)
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We present Value Elimination, a new algorithm for Bayesian Inference. Given the same variable ordering information, Value Elimination can achieve performance that is within a constant factor of variable elimination or recursive conditioning, and on some problems it can perform exponentially better, irrespective of the variable ordering used by these algorithms. Value Elimination
Compiling Bayesian Networks with Local Structure
"... Recent work on compiling Bayesian networks has reduced the problem to that of factoring CNF encodings of these networks, providing an expressive framework for exploiting local structure. For networks that have local structure, large CPTs, yet no excessive determinism, the quality of the CNF encoding ..."
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Cited by 42 (7 self)
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Recent work on compiling Bayesian networks has reduced the problem to that of factoring CNF encodings of these networks, providing an expressive framework for exploiting local structure. For networks that have local structure, large CPTs, yet no excessive determinism, the quality of the CNF encodings and the amount of local structure they capture can have a significant effect on both the offline compile time and online inference time. We examine the encoding of such Bayesian networks in this paper and report on new findings that allow us to significantly scale this compilation approach. In particular, we obtain order–of–magnitude improvements in compile time, compile some networks successfully for the first time, and obtain orders– of–magnitude improvements in online inference for some networks with local structure, as compared to baseline jointree inference, which does not exploit local structure.
Heuristics for fast exact model counting
 In Proc. 8th International Conference on Theory and Applications of Satisfiability Testing
, 2005
"... Abstract. An important extension of satisfiability testing is modelcounting, a task that corresponds to problems such as probabilistic reasoning and computing the permanent of a Boolean matrix. We recently introduced Cachet, an exact modelcounting algorithm that combines formula caching, clause le ..."
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Cited by 29 (2 self)
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Abstract. An important extension of satisfiability testing is modelcounting, a task that corresponds to problems such as probabilistic reasoning and computing the permanent of a Boolean matrix. We recently introduced Cachet, an exact modelcounting algorithm that combines formula caching, clause learning, and component analysis. This paper reports on experiments with various techniques for improving the performance of Cachet, including componentselection strategies, variableselection branching heuristics, randomization, backtracking schemes, and crosscomponent implications. The result of this work is a highlytuned version of Cachet, the first (and currently, only) system able to exactly determine the marginal probabilities of variables in random 3SAT formulas with 150+ variables. We use this to discover an interesting property of random formulas that does not seem to have been previously observed. 1
Performing bayesian inference by weighted model counting
 In Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2005
"... Over the past decade general satisfiability testing algorithms have proven to be surprisingly effective at solving a wide variety of constraint satisfaction problem, such as planning and scheduling (Kautz and Selman 2003). Solving such NPcomplete tasks by “compilation to SAT ” has turned out to be a ..."
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Cited by 28 (0 self)
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Over the past decade general satisfiability testing algorithms have proven to be surprisingly effective at solving a wide variety of constraint satisfaction problem, such as planning and scheduling (Kautz and Selman 2003). Solving such NPcomplete tasks by “compilation to SAT ” has turned out to be an approach that is of both practical and theoretical interest. Recently, (Sang et al. 2004) have shown that state of the art SAT algorithms can be efficiently extended to the harder task of counting the number of models (satisfying assignments) of a formula, by employing a technique called component caching. This paper begins to investigate the question of whether “compilation to modelcounting ” could be a practical technique for solving realworld #Pcomplete problems, in particular Bayesian inference. We describe an efficient translation from Bayesian networks to weighted model counting, extend the best modelcounting algorithms to weighted model counting, develop an efficient method for computing all marginals in a single counting pass, and evaluate the approach on computationally challenging reasoning problems.
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 24 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,
On probabilistic inference by weighted model counting
 Artificial Intelligence
"... A recent and effective approach to probabilistic inference calls for reducing the problem to one of weighted model counting (WMC) on a propositional knowledge base. Specifically, the approach calls for encoding the probabilistic model, typically a Bayesian network, as a propositional knowledge base ..."
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Cited by 22 (0 self)
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A recent and effective approach to probabilistic inference calls for reducing the problem to one of weighted model counting (WMC) on a propositional knowledge base. Specifically, the approach calls for encoding the probabilistic model, typically a Bayesian network, as a propositional knowledge base in conjunctive normal form (CNF) with weights associated to each model according to the network parameters. Given this CNF, computing the probability of some evidence becomes a matter of summing the weights of all CNF models consistent with the evidence. A number of variations on this approach have appeared in the literature recently, that vary across three orthogonal dimensions. The first dimension concerns the specific encoding used to convert a Bayesian network into a CNF. The second dimensions relates to whether weighted model counting is performed using a search algorithm on the CNF, or by compiling the CNF into a structure that renders WMC a polytime operation in the size of the compiled structure. The third dimension deals with the specific properties of network parameters (local structure) which are captured in the CNF encoding. In this paper, we discuss recent work in this area across the above three dimensions, and demonstrate empirically its practical importance in significantly expanding the reach of exact probabilistic inference. We restrict our discussion to exact inference and model counting, even though other proposals have been extended for approximate inference and approximate model counting.
DPLL with a trace: From SAT to knowledge compilation
 IJCAI05
, 2005
"... We show that the trace of an exhaustive DPLL search can be viewed as a compilation of the propositional theory. With different constraints imposed or lifted on the DPLL algorithm, this compilation will belong to the language of dDNNF, FBDD, and OBDD, respectively. These languages are decreasingly s ..."
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Cited by 21 (2 self)
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We show that the trace of an exhaustive DPLL search can be viewed as a compilation of the propositional theory. With different constraints imposed or lifted on the DPLL algorithm, this compilation will belong to the language of dDNNF, FBDD, and OBDD, respectively. These languages are decreasingly succinct, yet increasingly tractable, supporting such polynomialtime queries as model counting and equivalence testing. Our contribution is thus twofold. First, we provide a uniform framework, supported by empirical evaluations, for compiling knowledge into various languages of interest. Second, we show that given a particular variant of DPLL, by identifying the language membership of its traces, one gains a fundamental understanding of the intrinsic complexity and computational power of the search algorithm itself. As interesting examples, we unveil the “hidden power” of several recent model counters, point to one of their potential limitations, and identify a key limitation of DPLLbased procedures in general.
Nearuniform sampling of combinatorial spaces using xor constraints
 In NIPS. 2007
"... We propose a new technique for sampling the solutions of combinatorial problems in a nearuniform manner. We focus on problems specified as a Boolean formula, i.e., on SAT instances. Sampling for SAT problems has been shown to have interesting connections with probabilistic reasoning, making practic ..."
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Cited by 16 (3 self)
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We propose a new technique for sampling the solutions of combinatorial problems in a nearuniform manner. We focus on problems specified as a Boolean formula, i.e., on SAT instances. Sampling for SAT problems has been shown to have interesting connections with probabilistic reasoning, making practical sampling algorithms for SAT highly desirable. The best current approaches are based on Markov Chain Monte Carlo methods, which have some practical limitations. Our approach exploits combinatorial properties of random parity (XOR) constraints to prune away solutions nearuniformly. The final sample is identified amongst the remaining ones using a stateoftheart SAT solver. The resulting sampling distribution is provably arbitrarily close to uniform. Our experiments show that our technique achieves a significantly better sampling quality than the best alternative. 1
Leveraging belief propagation, backtrack search, and statistics for model counting
"... Abstract. We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different tradeoffs between efficiency, bound quality, and correctness guarantee ..."
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Cited by 14 (5 self)
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Abstract. We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different tradeoffs between efficiency, bound quality, and correctness guarantee. For lower bounds, we use a recent framework for probabilistic correctness guarantees, and exploit message passing techniques for marginal probability estimation, namely, variations of Belief Propagation (BP). Our results suggest that BP provides useful information even on structured loopy formulas. For upper bounds, we perform multiple runs of the MiniSat SAT solver with a minor modification, and obtain statistical bounds on the model count based on the observation that the distribution of a certain quantity of interest is often very close to the normal distribution. Our experiments demonstrate that our model counters based on these two ideas, BPCount and MiniCount, can provide very good bounds in time significantly less than alternative approaches. 1