Abstract
We show that a conjunctive normal form (CNF) formula F is unsatisfiable iff there is a set of points of the Boolean space that is stable with respect to F. So testing the satisfiability of a CNF formula reduces to looking for a stable set of points (SSP). We give some properties of SSPs and describe a simple algorithm for constructing an SSP for a CNF formula. Building an SSP can be viewed as a “natural” way of search space traversal. This naturalness of search space examination allows one to make use of the regularity of CNF formulas to be checked for satisfiability. We illustrate this point by showing that if a CNF F formula is symmetric with respect to a group of permutations, it is very easy to make use of this symmetry when constructing an SSP. As an example, we show that the unsatisfiability of pigeon-hole CNF formulas can be proven by examining only a set of points whose size is quadratic in the number of holes.
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References
C.A. Brown, L. Finkelstein, P.W. Purdom. Backtrack searching in the presence of symmetry. In “Applied algebra, algebraic algorithms and error correcting codes”. Sixth international conference, P. 99–110. Springer-Verlag, 1988.
V. Chvatal, E. Szmeredi. Many hard examples for resolution. J. of the ACM,vol. 35, No 4, pp. 759–568.
W. Cook, C.R. Coullard, G. Turan. On the complexity of cutting planes proofs. Discrete Applied Mathematics, 18, 1987, 25–38.
J. Crawford, M. Ginsberg, E. Luks, A. Roy. Symmetry breaking predicates for search problems. Fifth International Conference on Principles of Knowledge Representation and Reasoning (KR’96).
M. Davis, G. Logemann, D. Loveland. A Machine program for theorem proving. Communications of the ACM.-1962.-V. 5.-P. 394–397.
E. Goldberg. Proving unsatisfiability of CNFs locally. Proceedings of LICS 2001 Workshop on Theory and Applications of Satisfiability Testing.
A. Haken. The intractability of resolution. Theor. Comput. Sci. 39 (1985), 297–308.
B. Krishnamurthy. Short proofs for tricky formulas. Acta Informatica 22 (1985) 253–275.
D. Mitchell, B. Selman, and H.J. Levesque. Hard and easy distributions of SAT problems. Proceedings AAAI-92, San Jose,CA, 459–465.
M. Moskewicz, C. Madigan, Y. Zhao, L. Zhang, S. Malik. Chaff: Engineering an Efficient SAT Solver. Proceedings of DAC-2001.
C. Papadimitriou. On selecting a satisfying truth assignment. Proceedings of FOC-91.
A. Roy. Symmetry breaking and fault tolerance in Boolean satisfiability. PhD thesis. Downloadable from http://www.cs.uoregon.edu/~aroy/
B. Selman, H. Kautz, B. Cohen. Noise strategies for improving local search. Proceedings of AAAI-94.
I. Shlyakhter. Generating effective symmetry breaking predicates for search problems. Proceedings of LICS 2001 Workshop on Theory and Applications of Satisfiability Testing
A. Urquhart. The symmetry rule in propositional logic. Discrete Applied Mathematics 96–97(1999):177–193,1999.
H. Wong-Toi. Private communication.
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Goldberg, E. (2002). Testing Satisfiability of CNF Formulas by Computing a Stable Set of Points. In: Voronkov, A. (eds) Automated Deduction—CADE-18. CADE 2002. Lecture Notes in Computer Science(), vol 2392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45620-1_15
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DOI: https://doi.org/10.1007/3-540-45620-1_15
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