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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.
The TPTP Problem Library
, 1999
"... This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for buildin ..."
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Cited by 100 (6 self)
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This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for building the library; ffl a discussion of the inadequacies of previous problem collections, and how these have been resolved in the TPTP; ffl a description of the library structure, including overview information; ffl descriptions of supplementary utility programs; ffl guidelines for obtaining and using the library; Contents 1 Introduction 2 1.1 Previous Problem Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 What is Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Inside the TPTP 6 2.1 The TPTP Domain Structure . . . . . . . . . . . . . . . . . . . . . ...
Search in a Small World
, 1998
"... In a graph with a "small world" topology, nodes are highly clustered yet the path length between them is small. Such a topology can make search problems very difficulty since local decisions quickly propagate globally. We show that graphs associated with many different search problems have a s ..."
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Cited by 97 (13 self)
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In a graph with a "small world" topology, nodes are highly clustered yet the path length between them is small. Such a topology can make search problems very difficulty since local decisions quickly propagate globally. We show that graphs associated with many different search problems have a small world topology, and that the cost of solving search problems with such a topology can have a heavytailed distribution.
Problem Structure in the Presence of Perturbations
 In Proceedings of the 14th National Conference on AI
, 1997
"... Recent progress on search and reasoning procedures has been driven by experimentation on computationally hard problem instances. Hard random problem distributions are an important source of such instances. Challenge problems from the area of finite algebra have also stimulated research on searc ..."
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Cited by 70 (17 self)
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Recent progress on search and reasoning procedures has been driven by experimentation on computationally hard problem instances. Hard random problem distributions are an important source of such instances. Challenge problems from the area of finite algebra have also stimulated research on search and reasoning procedures. Nevertheless, the relation of such problems to practical applications is somewhat unclear. Realistic problem instances clearly have more structure than the random problem instances, but, on the other hand, they are not as regular as the structured mathematical problems. We propose a new benchmark domain that bridges the gap between the purely random instances and the highly structured problems, by introducing perturbations into a structured domain. We will show how to obtain interesting search problems in this manner, and how such problems can be used to study the robustness of search control mechanisms. Our experiments demonstrate that the performan...
PSATO: a Distributed Propositional Prover and Its Application to Quasigroup Problems
 Journal of Symbolic Computation
, 1996
"... This paper shows a way of using such resources to solve hard problems. ..."
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Cited by 68 (4 self)
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This paper shows a way of using such resources to solve hard problems.
Kodkod: A relational model finder
 In Tools and Algorithms for Construction and Analysis of Systems (TACAS
, 2007
"... Abstract. The key design challenges in the construction of a SATbased relational model finder are described, and novel techniques are proposed to address them. An efficient model finder must have a mechanism for specifying partial solutions, an effective symmetry detection and breaking scheme, and ..."
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Cited by 60 (4 self)
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Abstract. The key design challenges in the construction of a SATbased relational model finder are described, and novel techniques are proposed to address them. An efficient model finder must have a mechanism for specifying partial solutions, an effective symmetry detection and breaking scheme, and an economical translation from relational to boolean logic. These desiderata are addressed with three new techniques: a symmetry detection algorithm that works in the presence of partial solutions, a sparsematrix representation of relations, and a compact representation of boolean formulas inspired by boolean expression diagrams and reduced boolean circuits. The presented techniques have been implemented and evaluated, with promising results. 1
Implementing the DavisPutnam Method
 Journal of Automated Reasoning
, 2000
"... The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the DavisPutnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the DavisPutnam method efficiently usin ..."
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Cited by 53 (3 self)
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The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the DavisPutnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the DavisPutnam method efficiently using the trie data structure for propositional clauses. A new technique of indexing only the first and last literals of clauses yields a unit propagation procedure whose complexity is sublinear to the number of occurrences of the variable in the input. We also show that the DavisPutnam method can work better when unit subsumption is not used. We illustrate the performance of our programs on some quasigroup problems. The efficiency of our programs has enabled us to solve some open quasigroup problems.
Ordered Binary Decision Diagrams and the DavisPutnam Procedure
 IN PROC. OF THE 1ST INTERNATIONAL CONFERENCE ON CONSTRAINTS IN COMPUTATIONAL LOGICS
, 1994
"... We compare two prominent decision procedures for propositional logic: Ordered Binary Decision Diagrams (obdds) and the DavisPutnam procedure. Experimental results indicate that the DavisPutnam procedure outperforms obdds in hard constraintsatisfaction problems, while obdds are clearly superior for ..."
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Cited by 44 (1 self)
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We compare two prominent decision procedures for propositional logic: Ordered Binary Decision Diagrams (obdds) and the DavisPutnam procedure. Experimental results indicate that the DavisPutnam procedure outperforms obdds in hard constraintsatisfaction problems, while obdds are clearly superior for Boolean functional equivalence problems from the circuit domain, and, in general, problems that require the schematization of a large number of solutions that share a common structure. The two methods illustrate the different and often complementary strengths of constraintoriented and searchoriented procedures.
Implementing the DavisPutnam Algorithm by Tries
 ARTIFICIAL INTELLIGENCE CENTER, SRI INTERNATIONAL, MENLO
, 1994
"... The DavisPutnam method is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the DavisPutnam method efficiently using the trie data structure for propositional clauses by presenting seven implementations of the method. We prop ..."
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Cited by 37 (7 self)
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The DavisPutnam method is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the DavisPutnam method efficiently using the trie data structure for propositional clauses by presenting seven implementations of the method. We propose a new technique for implementing unit propagation whose complexity is sublinear to the number of occurrences of the variable in the input. We present the performance of our programs on some quasigroup problems. The efficiency of our programs allowed us to solve some open quasigroup problems.
Compiling Problem Specifications into SAT
, 2001
"... We present a compiler that translates a problem specification into a propositional satisfiability test (SAT). Problems are specified in a logicbased language, called npspec, which allows the definition of complex problems in a highly declarative way, and whose expressive power is such to captu ..."
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Cited by 36 (10 self)
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We present a compiler that translates a problem specification into a propositional satisfiability test (SAT). Problems are specified in a logicbased language, called npspec, which allows the definition of complex problems in a highly declarative way, and whose expressive power is such to capture exactly all problems which belong to the complexity class NP. The target SAT instance is solved using any of the various stateoftheart solvers available from the community. The system obtained is an executable specification language for all NP problems which shows interesting computational properties. The performances of the system have been tested on a few classical problems, namely graph coloring, Hamiltonian cycle, and jobshop scheduling. c flSpringerVerlag. To appear on the Proceedings of the European Symposium On Programming (ESOP 2001) Genova, Italy, April 26, 2001 Lecture Notes in Computer Science, SpringerVerlag, 2001. ftp://ftp.dis.uniroma1.it/pub/ai/papers/cadoscha01.ps.gz 1