Formula dissection: A parallel algorithm for constraint satisfaction

Many well-known problems in Artificial Intelligence can be formulated in terms of systems of constraints. The problem of testing the satisfiability of propositional formulae (SAT) is of special importance due to its numerous applications in theoretical computer science and Artificial Intelligence. A brute-force algorithm for SAT will have exponential time complexity O(2^n), where n is the number of Boolean variables of the formula. Unfortunately, more sophisticated approaches such as resolution result in similar performances in the worst case. In this paper, we present a simple and relatively efficient parallel divide-and-conquer method to solve various subclasses of SAT. The dissection stage of the parallel algorithm splits the original formula into smaller subformulae with only a bounded number of interacting variables. In particular, we derive a parallel algorithm for the class of formulae whose corresponding graph representation is planar. Our parallel algorithm for planar 3-SAT has the worst-case performance of 2^O^(^n^) on a PRAM (parallel random access model) computer. Applications of our method to constraint satisfaction problems are discussed.

[1]  Frank Harary,et al.  Graph Theory , 2016 .

[2]  Krzysztof R. Apt,et al.  Principles of constraint programming , 2003 .

[3]  Joseph JáJá,et al.  Parallel Algorithms in Graph Theory: Planarity Testing , 1982, SIAM J. Comput..

[4]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

[5]  Armin Haken The intractability of resolution (complexity) , 1984 .

[6]  Jack Minker,et al.  Graph dissection techniques for vlsi and algorithms , 1987 .

[7]  Raimund Seidel,et al.  A New Method for Solving Constraint Satisfaction Problems , 1981, IJCAI.

[8]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[9]  Gerald J. Sussman,et al.  Forward Reasoning and Dependency-Directed Backtracking in a System for Computer-Aided Circuit Analysis , 1976, Artif. Intell..

[10]  Steven Fortune,et al.  Parallelism in random access machines , 1978, STOC.

[11]  Victor Y. Pan,et al.  Efficient parallel solution of linear systems , 1985, STOC '85.

[12]  David L. Waltz,et al.  Understanding Line drawings of Scenes with Shadows , 1975 .

[13]  Hilary Putnam,et al.  A Computing Procedure for Quantification Theory , 1960, JACM.

[14]  Alan K. Mackworth Consistency in Networks of Relations , 1977, Artif. Intell..

[15]  Azriel Rosenfeld,et al.  Scene Labeling by Relaxation Operations , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[16]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..

[17]  Alasdair Urquhart,et al.  Formal Languages]: Mathematical Logic--mechanical theorem proving , 2022 .

[18]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[19]  H. Djidjev On the Problem of Partitioning Planar Graphs , 1982 .

[20]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[21]  Edward P. K. Tsang,et al.  Foundations of constraint satisfaction , 1993, Computation in cognitive science.

[22]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[23]  Stuart C. Shapiro,et al.  Using Active Connection Graphs for Reasoning with Recursive Rules , 1981, IJCAI.

[24]  Eugene C. Freuder,et al.  The Complexity of Some Polynomial Network Consistency Algorithms for Constraint Satisfaction Problems , 1985, Artif. Intell..

[25]  James R. Slagle,et al.  Using Rewriting Rules for Connection Graphs to Prove Theorems , 1979, Artif. Intell..

[26]  Rina Dechter,et al.  Constraint Processing , 1995, Lecture Notes in Computer Science.

[27]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1984, STOC '84.

[28]  Simon Kasif,et al.  On the Parallel Complexity of Some Constraint Satisfaction Problems , 1986, AAAI.

[29]  Jeffrey D Ullma Computational Aspects of VLSI , 1984 .

[30]  Robert A. Kowalski,et al.  A Proof Procedure Using Connection Graphs , 1975, JACM.

[31]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[32]  Patrick Henry Winston,et al.  The psychology of computer vision , 1976, Pattern Recognit..

[33]  Christos H. Papadimitriou,et al.  The complexity of recognizing polyhedral scenes , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[34]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[35]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[36]  Charles E. Leiserson,et al.  Fat-trees: Universal networks for hardware-efficient supercomputing , 1985, IEEE Transactions on Computers.