Eli Ben-Sasson
Abstract
The widthof a Resolution proof is defined to be the maximal number of literals in any clause of the proof. In this paper, we relate proof width to proof length (=size), in both general Resolution, and its tree-like variant. The following consequences of these relations reveal width as a crucial “resource” of Resolution proofs.
In one direction, the relations allow us to give simple, unified proofs for almost all known exponential lower bounds on size of resolution proofs, as well as several interesting new ones. They all follow from width lower bounds, and we show how these follow from natural expansion property of clauses of the input tautology.
In the other direction, the width-size relations naturally suggest a simple dynamic programming procedure for automated theorem proving—one which simply searches for small width proofs. This relation guarantees that the runnuing time (and thus the size of the produced proof) is at most quasi-polynomial in the smallest tree-like proof. This algorithm is never much worse than any of the recursive automated provers (such as DLL) used in practice. In contrast, we present a family of tautologies on which it is exponentially faster.
- BEAME, P., KARP, R., PITASSI, T., AND SAKS, M. 2000. On the complexity of unsatisfiability proofs for random k-CNF formulas. Submitted.Google Scholar
- BEAME, P., AND PITASSI, T. 1996. Simplified and improved resolution lower bounds. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, Vt., Oct.). IEEE Computer Society Press, Los Alamitos, Calif., pp. 274-282. Google ScholarDigital Library
- BEN-SASSON, E., IMPAGLIAZZO, R., AND WIGDERSON, A. 2000. Optimal separation of tree-like and general resolution. In Electronic Colloquium on Computational Complexity Report Series. Tech. Rep. TR00-005.Google Scholar
- BONET,M.L.,ESTEBAN,J.L.,GALESI, N., AND JOHANNSEN, J. 1998. On the relative complexity of resolution refinements and cutting planes proof systems. In Electronic Colloquium on Computational Complexity Report Series. Tech. Rep. TR98-035. Google ScholarDigital Library
- BONET,M.L.,AND GALESI, N. 1999. A study of proof search algorithms for resolution and polynomial calculus. In Proceedings of the 40th Symposium of Foundations of Computer Science. IEEE Computer Society, Los Alamitos, Calif., pp. 422-431. Google ScholarDigital Library
- BUSS, S., AND PITASSI, T. 1997. Resolution and the weak pigeonhole principle. In Proceedings of the Conference for Computer Science Logic. Google ScholarDigital Library
- BUSS,S.R.,AND TURAN, G. 1988. Resolution proof of generalized pigeonhole principles. Theoret. Comput. Sci. 62, 311-317. Google ScholarDigital Library
- CELONI, R., PAUL,W.J.,AND TARJAN, R. E. 1977. Space bounds for a game on graphs. Math. Syst. Theory 10, 239-251.Google Scholar
- CHVATAL, V., AND SZEMERE 'DI, E. 1988. Many hard examples for resolution. J. ACM 35, 4 (Oct.), 759-768. Google ScholarDigital Library
- CLEGG, M., EDMONDS, J., AND IMPAGLIAZZO, R. 1996. Using the Groebner basis algorithm to find proofs of unsatisfiability. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (Philadelphia, Pa., May 22-24). ACM, New York, pp. 174-183. Google ScholarDigital Library
- COOK,S.A.,AND RECKHOW, R. 1979. The relative efficiency of propositional proof systems. J. Symb. Logic 44, 36-50.Google ScholarCross Ref
- DAVIS, M., LOGEMANN, G., AND LOVELAND, D. 1962. A machine program for theorem-proving. Commun. ACM 5, 7 (July), 394-397. Google ScholarDigital Library
- GALIL, Z. 1977. On resolution with clauses of bounded size. SIAM J. Comput. 6, 444-459.Google ScholarDigital Library
- HAKEN, A. 1985. The intractibility of resolution. Theoret. Comput. Sci. 39, 297-308.Google ScholarCross Ref
- IMPAGLIAZZO, R., PUDLAK, P., AND SGALL, J. 1997. Lower bounds for the polynomial-calculus and the groebner basis algorithm. Tech. Rep. TR97-042. Found at Electronic Colloqium on Computational Complexity, Reports Series 1997. Available at http://www.eccc.uni-trier.de/eccc/.Google Scholar
- PIPPENGER PEBBLING, N. 1980. IBM Reserach Report RC8258. Appeared in Proceedings of the 5th IBM Symposium on Mathematical Foundations of Computer Science, Japan.Google Scholar
- RAZ, R., AND MCKENZIE, P. 1999. Separation of the monotone NC hierarchy. Combinatorica 19,3, 403-435.Google ScholarDigital Library
- RAZBOROV, A. A. 1995. Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic. Izv. RAN 59, 1, 201-222.Google Scholar
- RAZBOROV,A.A.,AND RUDICH, S. 1994. Natural proofs. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing (Montreal, Que., Canada, May 23-25). ACM, New York, pp. 204-213. Google ScholarDigital Library
- RAZBOROV,A.A.,WIGDERSON, A., AND YAO, A. 1997. Read-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (El Paso, Tex., May 4-6). ACM, New York, pp. 739-748. Google ScholarDigital Library
- TSEITIN, G. S. 1968. On the complexity of derivation in propositional calculus. In Studies in Constructive Mathematics and Mathematical Logic, Part 2. Consultants Bureau, New York-London, pp. 115-125.Google Scholar
- URQUHART, A. 1987. Hard examples for resolution. J. ACM 34, 1 (Jan.), 209-219. Google ScholarDigital Library
- URQUHART, A. 1995. The complexity of propositional proofs. Bull. Symb. Logic 1, 4, 425-467.Google ScholarCross Ref
Index Terms
- Short proofs are narrow—resolution made simple
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Reviews
Ralph Walter Wilkerson
The width of a resolution proof is defined as the maximal number of literals in any clause of the proof. The authors argue that width as an important attribute by which to study resolution proofs. In particular, this concept is related to the length of such proofs. An immediate application of this work is a simplification and unification of most known exponential lower bounds on resolution proof lengths. Other applications of these methods are also given as they relate to automated theorem provers.
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