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81
A New Kind of Science
, 2002
"... “Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplit ..."
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Cited by 502 (0 self)
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“Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplitudes you told me about, they’re so complicated and absurd, what makes you think those are right? Maybe they aren’t right. ’ Such remarks are obvious and are perfectly clear to anybody who is working on this problem. It does not do any good to point this out.” —Richard Feynman [1, p.161]
HeavyTailed Phenomena in Satisfiability and Constraint Satisfaction Problems
 J. of Autom. Reasoning
, 2000
"... Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by ver ..."
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Cited by 148 (26 self)
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Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by very long tails or “heavy tails”. We will show that these distributions are best characterized by a general class of distributions that can have infinite moments (i.e., an infinite mean, variance, etc.). Such nonstandard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We also show how random restarts can effectively eliminate heavytailed behavior. Furthermore, for harder problem instances, we observe long tails on the lefthand side of the distribution, which is indicative of a nonnegligible fraction of relatively short, successful runs. A rapid restart strategy eliminates heavytailed behavior and takes advantage of short runs, significantly reducing expected solution time. We demonstrate speedups of up to two orders of magnitude on SAT and CSP encodings of hard problems in planning, scheduling, and circuit synthesis. Key words: satisfiability, constraint satisfaction, heavy tails, backtracking 1.
Automatic SATCompilation of Planning Problems
 IJCAI97
, 1997
"... Recent work by Kautz et al. provides tantalizing evidence that large, classical planning problems may be efficiently solved by translating them into propositional satisfiability problems, using stochastic search techniques, and translating the resulting truth assignments back into plans for the ..."
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Cited by 121 (10 self)
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Recent work by Kautz et al. provides tantalizing evidence that large, classical planning problems may be efficiently solved by translating them into propositional satisfiability problems, using stochastic search techniques, and translating the resulting truth assignments back into plans for the original problems. We explore the space of such transformations, providing a simple framework that generates eight major encodings (generated by selecting one of four action representations and one of two frame axioms) and a number of subsidiary ones. We describe a fullyimplemented compiler that can generate each of these encodings, and we test the compiler on a suite of STRIPS planning problems in order to determine which encodings have the best properties.
Ten challenges in propositional reasoning and search
, 1997
"... The past several years have seen much progress in the area of propositional reasoning and satisfiability testing. There is a growing consensus by researchers on the key technical challenges that need to be addressed in order to maintain this momentum. This paper outlines concrete technical challenge ..."
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Cited by 81 (3 self)
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The past several years have seen much progress in the area of propositional reasoning and satisfiability testing. There is a growing consensus by researchers on the key technical challenges that need to be addressed in order to maintain this momentum. This paper outlines concrete technical challenges in the core areas of systematic search, stochastic search, problem encodings, and criteria for evaluating progress in this area. 1
The LPSAT Engine & its Application to Resource Planning
, 1999
"... Compilation to boolean satisfiability has become a powerful paradigm for solving AI problems. However, domains... ..."
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Cited by 70 (0 self)
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Compilation to boolean satisfiability has become a powerful paradigm for solving AI problems. However, domains...
EFFICIENT ALGORITHMS FOR CLAUSELEARNING SAT SOLVERS
, 2004
"... Boolean satisfiability (SAT) is NPcomplete. No known algorithm for SAT is of polynomial time complexity. Yet, many of the SAT instances generated as a means of solving realworld electronic design automation problems are simple enough, structurally, that modern solvers can decide them efficiently. ..."
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Cited by 58 (0 self)
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Boolean satisfiability (SAT) is NPcomplete. No known algorithm for SAT is of polynomial time complexity. Yet, many of the SAT instances generated as a means of solving realworld electronic design automation problems are simple enough, structurally, that modern solvers can decide them efficiently. Consequently, SAT solvers are widely used in industry for logic verification. The most robust solver algorithms are poorly understood and only vaguely described in the literature of the field. We refine these algorithms, and present them clearly. We introduce several new techniques for Boolean constraint propagation that substantially improve solver efficiency. We explain why literal count decision strategies succeed, and on that basis, we introduce a new decision strategy that outperforms the state of the art. The culmination of this work is the most powerful SAT solver publically available.
PartitionBased Logical Reasoning for FirstOrder and Propositional Theories
 Artificial Intelligence
, 2000
"... In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with ..."
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Cited by 51 (8 self)
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In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward messagepassing and using backward messagepassing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our messagepassing ...
Enhancing Davis Putnam with Extended Binary Clause Reasoning
, 2002
"... The backtracking based Davis Putnam (DPLL) procedure remains the dominant method for deciding the satisfiability of a CNF formula. In recent years there has been much work on improving the basic procedure by adding features like improved heuristics and data structures, intelligent backtracking, ..."
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Cited by 47 (4 self)
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The backtracking based Davis Putnam (DPLL) procedure remains the dominant method for deciding the satisfiability of a CNF formula. In recent years there has been much work on improving the basic procedure by adding features like improved heuristics and data structures, intelligent backtracking, clause learning, etc. Reasoning with binary clauses in DPLL has been a much discussed possibility for achieving improved performance, but to date solvers based on this idea have not been competitive with the best unit propagation based DPLL solvers. In this paper we experiment with a DPLL solver called 2CLS+EQ that makes more extensive use of binary clause reasoning than has been tried before. The results are very encouraging2CLS+EQ is competitive with the very best DPLL solvers. The techniques it uses also open up a number of other possibilities for increasing our ability to solve SAT problems.
Solving Difficult Instances of Boolean Satisfiability in the Presence of Symmetry
, 2002
"... Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large siz ..."
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Cited by 44 (17 self)
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Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large size, but are still solved in minutes. Yet, small and difficult SAT instances must exist if P##NP. To this end, our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from Very Large Scale Integration (VLSI) routing problems. With an efficient implementation to solve the graph automorphism problem [39, 50, 51], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.
Backbone Fragility and the Local Search Cost Peak
 Journal of Artificial Intelligence Research
, 2000
"... The local search algorithm WSat is one of the most successful algorithms for solving the satisfiability (SAT) problem. It is notably e#ective at solving hard Random 3SAT instances near the socalled `satisfiability threshold', but still shows a peak in search cost near the threshold and large va ..."
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Cited by 40 (3 self)
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The local search algorithm WSat is one of the most successful algorithms for solving the satisfiability (SAT) problem. It is notably e#ective at solving hard Random 3SAT instances near the socalled `satisfiability threshold', but still shows a peak in search cost near the threshold and large variations in cost over di#erent instances. We make a number of significant contributions to the analysis of WSat on highcost random instances, using the recentlyintroduced concept of the backbone of a SAT instance. The backbone is the set of literals which are entailed by an instance. We find that the number of solutions predicts the cost well for smallbackbone instances but is much less relevant for the largebackbone instances which appear near the threshold and dominate in the overconstrained region. We show a very strong correlation between search cost and the Hamming distance to the nearest solution early in WSat's search. This pattern leads us to introduce a measure of the ba...