Notre Dame Journal of Formal Logic

Simplified Lower Bounds for Propositional Proofs

Xudong Fu and Alasdair Urquhart

Abstract

This article presents a simplified proof of the result that bounded depth propositional proofs of the pigeonhole principle are exponentially large. The proof uses the new techniques for proving switching lemmas developed by Razborov and Beame. A similar result is also proved for some examples based on graphs.

Article information

Source
Notre Dame J. Formal Logic, Volume 37, Number 4 (1996), 523-544.

Dates
First available in Project Euclid: 16 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1040046140

Digital Object Identifier
doi:10.1305/ndjfl/1040046140

Mathematical Reviews number (MathSciNet)
MR1446227

Zentralblatt MATH identifier
0882.03052

Subjects
Primary: 03F20: Complexity of proofs
Secondary: 68Q25: Analysis of algorithms and problem complexity [See also 68W40]

Citation

Urquhart, Alasdair; Fu, Xudong. Simplified Lower Bounds for Propositional Proofs. Notre Dame J. Formal Logic 37 (1996), no. 4, 523--544. doi:10.1305/ndjfl/1040046140. https://projecteuclid.org/euclid.ndjfl/1040046140


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References

  • [1]Ajtai, M., ``The complexity of the pigeonhole principle," pp. 346--355 in Proceedings of the 29th Annual IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 1988.
  • [2]Beame, P. ``A switching lemma primer," preprint, 1993.
  • [3]Beame, P., R. Impagliazzo, J. Krají\ucek, T. Pitassi, P. Pudlák, and A. Woods, ``Exponential lower bounds for the pigeonhole principle," pp. 200--220 in Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, ACM Press, 1992.
  • [4]Bellantoni, S., T. Pitassi, and A. Urquhart, ``Approximation and small-depth Frege proofs," SIAM Journal of Computing, vol. 21 (1992), pp. 1161--1179.
  • [5]Chvátal, V., and E. Szemerédi, ``Many hard examples for resolution,'' Journal of the Association for Computing Machinery, vol. 35 (1988), pp. 759--768.
  • [6]Cook, S. A., Resolution Lower Bound for Complete Graph Clauses, manuscript, University of Toronto, Toronto, 1993.
  • [7]Furst, M., J. B. Saxe, and M. Sipser, ``Parity, circuits, and the polynomial-time hierarchy,'' pp. 260--270 in Proceedings of the 22nd Annual IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 1981.
  • [8]Frege, G., Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Nebert, Halle, 1879.
  • [9] Hå stad, J. T., Computational Limitations of Small-Depth Circuits, MIT Press, Cambridge, 1987.
  • [10]Krají\ucek, J., P. Pudlák, and A. Woods, ``Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle," Random Structures and Algorithms, vol. 7 (1995), pp. 15--39.
  • [11] Von Neumann, J., ``Zur Hilbertschen Beweistheorie," Mathematische Zeitschrift, vol. 26 (1926), pp. 1--46.
  • [12]Pitassi. T., P. Beame, and R. Impagliazzo, ``Exponential lower bounds for the pigeonhole principle," Computational Complexity, vol. 3 (1993), pp. 97--140.
  • [13]Razborov, A. A., ``Bounded arithmetic and lower bounds in Boolean complexity,'' pp. 344--386 in Feasible Mathematics II, edited by P. Clote and J. Remmel, Birkhäuser, Boston, 1995.
  • [14]Shoenfield, J., Mathematical Logic, Addison-Wesley, Reading, 1967.
  • [15]Tseitin, G. S., ``On the complexity of derivation in propositional calculus," pp. 115--125 in Studies in Constructive Mathematics and Mathematical Logic, Part 2, edited by A. O. Slisenko, 1970 (reprinted in Automation of Reasoning Vol. 2, edited by J. Siekmann and G. Wrightson, Springer-Verlag, New York, 1983, pp. 466--483).
  • [16]Urquhart, A., ``Hard examples for resolution,'' Journal of the Association for Computing Machinery, vol. 34 (1987), pp. 209--219.
  • [17]Yao, A., ``Separating the polynomial-time hierarchy by oracles,'' pp. 1--10 in Proceedings of the 26th Annual IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 1985.