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Polynomial size proofs of the propositional pigeonhole principle

Published online by Cambridge University Press:  12 March 2014

Samuel R. Buss
Samuel R. Buss*
Affiliation:
Mathematical Sciences Research Institute, Berkeley, California 94720
*
Department of Mathematics, University of California, Berkeley, California 94720
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Abstract

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 4 , December 1987 , pp. 916 - 927
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[1]Buss, S., Bounded arithmetic, Ph.D. Thesis, Princeton University, Princeton, New Jersey, 1985.Google Scholar
[2]Cook, S. and Reckhow, R., The relative efficiency of propositional proof systems, this Journal, vol. 44 (1979), pp. 36–50.Google Scholar
[3]Furst, M., Saxe, J. and Sipser, M., Parity, circuits and the polynomial-time hierarchy, IEEE Symposium on Foundation of Computing, vol. 22 (1981), pp. 260–270.Google Scholar
[4]Haken, A., The intractability of resolution, Theoretical Computer Science, vol. 39 (1985), pp. 297–308.CrossRefGoogle Scholar
[5]Paris, J. and Wilkie, A., Counting problems in bounded arithmetic, Methods in Mathematical Logic (Proceedings, Caracas, 1983), Lecture Notes in Mathematics, vol. 1130, Springer-Verlag, Berlin, 1985, pp. 317–340.CrossRefGoogle Scholar
[6]Savage, J. E., The complexity of computing, Wiley, New York, 1976.Google Scholar
[7]Statman, R., Complexity of derivations from quantifier-free Horn formulae, mechanical introduction of explicit definitions, and refinement of completeness theorems, Logic Colloquium '76 (Gandy, R. and Hyland, M., editors), North-Holland, Amsterdam, 1977, pp. 505–517.Google Scholar
[8]Tseĭtin, G. S., On the complexity of derivation in propositional calculus, Studies in constructive mathematics and mathematical logic. II (Slisenko, A. O., editor), Zapiski Nauchnykh Seminarov LOMI, vol. 8 (1968), pp. 234–259; English translation, Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, vol. 8, Consultants Bureau, New York, 1970, pp. 115–125.Google Scholar
[9]Wilkie, A., Talk presented at Logic Colloquium '84, Manchester, 1984.Google Scholar
[10]Woods, A., Some problems in logic and number theory and their connections, Ph.D. Thesis, Manchester University, Manchester, 1981.Google Scholar
[11]Urquhart, A., Hard examples for resolution, Journal of the Association for Computing Machinery, vol. 34 (1987), pp. 209–219.CrossRefGoogle Scholar
[12]Yao, A., Separating the polynomial-time hierarchy by oracles, IEEE Symposium on Foundations of Computing, vol. 26 (1985), pp. 1–10.Google Scholar