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58
Planning graphs and knowledge compilation
 In ICAPS
"... One of the major advances in classical planning has been the development of Graphplan. Graphplan builds a layered structure called the planning graph, and then searches this structure backwards for a plan. Modern SAT and CSP approaches also use the planning graph but replace the regression search by ..."
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One of the major advances in classical planning has been the development of Graphplan. Graphplan builds a layered structure called the planning graph, and then searches this structure backwards for a plan. Modern SAT and CSP approaches also use the planning graph but replace the regression search by a constraineddirected search. The planning graph uncovers implicit constraints in the problem that reduce the size of the search tree. Such constraints encode lower bounds on the number of time steps required for achieving the goal and account for the huge performance gap between Graphplan and its predecessors. Still, the form of local consistency underlying the construction of the planning graph is not well understood, being described by various authors as a limited form of negative binary resolution, kconsistency, or 2j consistency. In this paper, we aim to shed light on this issue by showing that the computation of the planning graph corresponds exactly to the iterative computation of prime implicates of size one and two over the logical encoding of the problem with the goals removed. The correspondence between planning graphs and a precise form of knowledge compilation provides a wellfounded basis for understanding and developing extensions of the planning graph to nonStrips settings, and suggests novel and effective forms of knowledge compilation in other contexts. We explore some of these extensions in this paper and relate planning graphs with bounded variable elimination algorithms as studied by Rina Dechter and others.
Extended Resolution Proofs for Symbolic SAT Solving with Quantification
 In Proc. 9th Intl. Conf. on Theory and Applications of Satisfiability Testing (SAT’06), Lecture Notes in Computer Science (LNCS
, 2006
"... Abstract. Symbolic SAT solving is an approach where the clauses of a CNF formula are represented using BDDs. These BDDs are then conjoined, and finally checking satisfiability is reduced to the question of whether the final BDD is identical to false. We present a method combining symbolic SAT solvin ..."
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Abstract. Symbolic SAT solving is an approach where the clauses of a CNF formula are represented using BDDs. These BDDs are then conjoined, and finally checking satisfiability is reduced to the question of whether the final BDD is identical to false. We present a method combining symbolic SAT solving with BDD quantification (variable elimination) and generation of extended resolution proofs. Proofs are fundamental to many applications, and our results allow the use of BDDs instead of—or in combination with—established proof generation techniques like clause learning. We have implemented a symbolic SAT solver with variable elimination that produces extended resolution proofs. We present details of our implementation, called EBDDRES, which is an extension of the system presented in [1], and also report on experimental results. 1
DPLL with Caching: A new algorithm for #SAT and Bayesian inference
 Electronic Colloquium in Computation Complexity
, 2003
"... Bayesian inference and counting satisfying assignments are important problems with numerous applications in probabilistic reasoning. In this paper, we show that plain old DPLL equipped with memoization can solve both of these problems with time complexity that is at least as good as all known algo ..."
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Cited by 10 (3 self)
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Bayesian inference and counting satisfying assignments are important problems with numerous applications in probabilistic reasoning. In this paper, we show that plain old DPLL equipped with memoization can solve both of these problems with time complexity that is at least as good as all known algorithms. Furthermore, DPLL with memoization achieves the best known timespace tradeoff. Although their worst case time complexity is no better, our DPLL based algorithms have the potential to achieve much better performance than known algorithms on problems which possess additional structure. Probabilistic models of real situations tend to have such additional structure.
Local search for unsatisfiability
 In Proceedings of SAT
, 2006
"... Abstract. Local search is widely applied to satisfiable SAT problems, and on some classes outperforms backtrack search. An intriguing challenge posed by Selman, Kautz and McAllester in 1997 is to use it instead to prove unsatisfiability. We investigate two distinct approaches. Firstly we apply stand ..."
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Cited by 9 (1 self)
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Abstract. Local search is widely applied to satisfiable SAT problems, and on some classes outperforms backtrack search. An intriguing challenge posed by Selman, Kautz and McAllester in 1997 is to use it instead to prove unsatisfiability. We investigate two distinct approaches. Firstly we apply standard local search to a reformulation of the problem, such that a solution to the reformulation corresponds to a refutation of the original problem. Secondly we design a greedy randomised resolution algorithm that will eventually discover proofs of any size while using bounded memory. We show experimentally that both approaches can refute some problems more quickly than backtrack search. 1
Solving #SAT and Bayesian inference with backtracking search
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 2009
"... Inference in Bayes Nets (BAYES) is an important problem with numerous applications in probabilistic reasoning. Counting the number of satisfying assignments of a propositional formula (#SAT) is a closely related problem of fundamental theoretical importance. Both these problems, and others, are memb ..."
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Cited by 8 (1 self)
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Inference in Bayes Nets (BAYES) is an important problem with numerous applications in probabilistic reasoning. Counting the number of satisfying assignments of a propositional formula (#SAT) is a closely related problem of fundamental theoretical importance. Both these problems, and others, are members of the class of sumofproducts (SUMPROD) problems. In this paper we show that standard backtracking search when augmented with a simple memoization scheme (caching) can solve any sumofproducts problem with time complexity that is at least as good any other stateoftheart exact algorithm, and that it can also achieve the best known timespace tradeoff. Furthermore, backtrackingâs ability to utilize more flexible variable orderings allows us to prove that it can achieve an exponential speedup over other standard algorithms for SUMPROD on some instances. The ideas presented here have been utilized in a number of solvers that have been applied to various types of sumofproduct problems. These systemâs have exploited the fact that backtracking can naturally exploit more of the problemâs structure to achieve improved performance on a range of problem instances. Empirical evidence of this performance gain has appeared in published works describing these solvers, and we provide references to these works.
Visualizing the internal structure of SAT instances (preliminary report
 In Proc. 7th Intl. Conf. on Theory and Applications of Satisfiability Testing (SAT 2004
, 2004
"... Abstract. Modern algorithms for the SAT problem reveal an almost tractable behavior on “realworld” instances. This is frequently contributed to the fact that these instances possess an internal “structure” that hard problem instances do not exhibit. However, little is known about this internal stru ..."
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Cited by 6 (2 self)
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Abstract. Modern algorithms for the SAT problem reveal an almost tractable behavior on “realworld” instances. This is frequently contributed to the fact that these instances possess an internal “structure” that hard problem instances do not exhibit. However, little is known about this internal structure. We therefore propose a visualization of the instance’s variable interaction graph (and of its dynamic change during a run of a SATsolver) as a first step of an empirical research program to analyze the problem structure. We present first results of such an analysis on instances of bounded model checking benchmark problems. 1
Modelling more realistic sat problems
 In Australian Joint Conference on Artificial Intelligence
, 2002
"... Abstract. The satisfiability problem is widely used in research on combinatorial search and for industrial applications such as verification and planning. Real world search problem benchmarks are not plentiful, yet understanding search algorithm behaviour in the real world domain is highly important ..."
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Cited by 5 (0 self)
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Abstract. The satisfiability problem is widely used in research on combinatorial search and for industrial applications such as verification and planning. Real world search problem benchmarks are not plentiful, yet understanding search algorithm behaviour in the real world domain is highly important. This work justifies and investigates a randomised satisfiability problem model with modular properties akin to those observed in real world search problem domains. The proposed problem model provides a reliable benchmark which highlights pitfalls and advantages with various satisfiability search algorithms. 1
Local Consistency and SATSolvers
"... Abstract. In this paper we show that the power of using kconsistency techniques in a constraint problem is precisely captured by using a particular inference rule, which we call positivehyperresolution, on the direct Boolean encoding of the CSP instance. We also show that current clauselearning S ..."
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Abstract. In this paper we show that the power of using kconsistency techniques in a constraint problem is precisely captured by using a particular inference rule, which we call positivehyperresolution, on the direct Boolean encoding of the CSP instance. We also show that current clauselearning SATsolvers will deduce any positivehyperresolvent of a fixed size from a given set of clauses in polynomial expected time. We combine these two results to show that, without being explicitly designed to do so, current clauselearning SATsolvers efficiently simulate kconsistency techniques, for all values of k. We then give some experimental results to show that this feature allows clauselearning SATsolvers to efficiently solve certain families of CSP instances which are challenging for conventional CP solvers. 1
DPvis  a tool to visualize the structure of SAT instances. SAT 2005 : international conference on theory and applications of satisfiability testing
 In Proc. of the 8th Intl. Conf. on Theory and Appl. of Satisfiability Testing (SAT
, 2005
"... Abstract. We present DPVIS, a Java tool to visualize the structure of SAT instances and runs of the DPLL (DavisPutnamLogemannLoveland) procedure. DPVIS uses advanced graph layout algorithms to display the problem’s internal structure arising from its variable dependency (interaction) graph. DPVIS ..."
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Cited by 4 (1 self)
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Abstract. We present DPVIS, a Java tool to visualize the structure of SAT instances and runs of the DPLL (DavisPutnamLogemannLoveland) procedure. DPVIS uses advanced graph layout algorithms to display the problem’s internal structure arising from its variable dependency (interaction) graph. DPVIS is also able to generate animations showing the dynamic change of a problem’s structure during a typical DPLL run. Besides implementing a simple variant of the DPLL algorithm on its own, DPVIS also features an interface to MiniSAT, a stateoftheart DPLL implementation. Using this interface, runs of MiniSAT can be visualized—including the generated search tree and the effects of clause learning. DPVIS is supposed to help in teaching the DPLL algorithm and in gaining new insights in the structure (and hardness) of SAT instances. 1
Projection Pushing Revisited
 In Proc of EDBT’04
, 2004
"... Abstract. The join operation, which combines tuples from multiple relations, is the most fundamental and, typically, the most expensive operation in database queries. The standard approach to joinquery optimization is cost based, which requires developing a cost model, assigning an estimated cost t ..."
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Abstract. The join operation, which combines tuples from multiple relations, is the most fundamental and, typically, the most expensive operation in database queries. The standard approach to joinquery optimization is cost based, which requires developing a cost model, assigning an estimated cost to each queryprocessing plan, and searching in the space of all plans for a plan of minimal cost. Two other approaches can be found in the databasetheory literature. The first approach, initially proposed by Chandra and Merlin, focused on minimizing the number of joins rather then on selecting an optimal join order. Unfortunately, this approach requires a homomorphism test, which itself is NPcomplete, and has not been pursued in practical query processing. The second, more recent, approach focuses on structural properties of the query in order to find a projectjoin order that will minimize the size of intermediate results during query evaluation. For example, it is known that for Boolean projectjoin queries a projectjoin order can be found such that the arity of intermediate results is the treewidth of the join graph plus one. In this paper we pursue the structuraloptimization approach, motivated by its success in the context of constraint satisfaction. We chose a setup in which the costbased approach is rather ineffective; we generate projectjoin queries with a large number of relations over databases with small relations. We show that a standard SQL planner (we use PostgreSQL) spends an exponential amount of time on generating plans for such queries, with rather dismal results in terms of performance. We then show how structural techniques, including projection pushing and join reordering, can yield exponential improvements in query execution time. Finally, we combine early projection and join reordering in an implementation of the bucketelimination method from constraint satisfaction to obtain another exponential improvement. 1