Exponential lower bounds for the pigeonhole principle
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
In this paper we prove an exponential lower bound on the size of bounded-depth Frege proofs for the pigeonhole principle (PHP). We also obtain an Ω(loglogn)-depth lower bound for any polynomial-sized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact complexity of the PHP, as S. Buss has constructed polynomial-size, logn-depth Frege proofs for the PHP. The main lemma in our proof can be viewed as a general Håstad-style Switching Lemma for restrictions that are partial matchings. Our lower bounds for the pigeonhole principle improve on previous superpolynomial lower bounds.
- M. Ajtai, The complexity of the pigeonhole principle, inProc. 29th Ann. IEEE Symp. Foundations of Computer Science, 1988, 346–355.
- M. Ajtai, 138-1 on finite structures,Annals of Pure and Applied Logic,24 (1983), 1–48.
- P. Beame, Lower bounds for recognizing small cliques on CRCW PRAM's,Discrete Applied Mathematics,29 (1990), 3–20.
- P. Beame, J. Håstad, Optimal bounds for decision problems on the CRCW PRAM,Journal of the ACM,36 (1989), 643–670.
- P. Beame, R. Implagiazzo, J. Krajíček, T. Pitassi, P. Pudlák, A. Woods, Exponential lower bounds for the pigeonhole principle,Proc. 24th Ann. ACM Symp. Theory of Computing 1992, 200–220.
- S. Bellantoni, T. Pitassi, A. Urquhart, Approximation and small-depth Frege proofs,SIAM Journal of Computing,21 (1992), 1161–1179.
- M. Bonet and S. Buss, The deduction rule and linear and near-linear proof simulations, preprint 1992.
- S. Buss, Polynomial size proofs of the propositional pigeonhole principle,Journal of Symbolic Logic,52 (1987), 916–927.
- S. Buss, Personal communication, 1993.
- S. A. Cook andR. Reckhow. The relative efficiency of propositional proof systems,Journal of Symbolic Logic,44 (1979), 36–50.
- M. Furst, J. Saxe, M. Sipser, Parity, circuits and the polynomial time hierarchy,Mathematical Systems Theory,17 (1984) 13–27.
- A. Haken, The intractability of Resolution,Theoretical Computer Science 39 (1985) 297–308.
- J. Håstad,Computational limitations of small-depth circuits, The MIT Press, Cambridge, Massachusetts, 1987.
- J. Krajíček, Lower bounds to the size of constant-depth propositional proofs, preprint (1991).
- J. Krajíček, P. Pudlák, A. Woods, Exponential lower bounds to the size of bounded-depth Frege proofs of the pigeonhole principle, preprint (1991).
- J. Lynch, A depth-size tradeoff for Boolean circuits with unbounded fan-in,Lecture Notes in Computer Science 223 (1986), 234–248.
- J. Paris, A. Wilkie, A. Woods, Provability of the pigeonhole principle and the existence of infinitely many primes,Journal of Symbolic Logic,53 Number 4 (1988).
- T. Pitassi, P. Beame, R. Impagliazzo, Exponential lower bounds for the pigeonhole principle, University of Toronto TR 257/91 (1991).
- G. S. Tseitin, On the complexity of derivation in the propositional calculus,Studies in Constructive Mathematics and Mathematical Logic, Part II, A.O. Slisenko, 1968.
- A. Urquhart, Hard examples for Resolution,JACM,34 (1987), 209–219.
- A. C. Yao, Separating the polynomial-time hierarchy by oracles,Proc. 26th Ann. IEEE Symp. Foundations of Computer Science, 1985, 1–10.
- Exponential lower bounds for the pigeonhole principle
Volume 3, Issue 2 , pp 97-140
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Complexity of propositional proof systems
- lower bounds
- Industry Sectors
- Author Affiliations
- 1. Department of Computer Science, University of California at San Diego, La Jolla, CA
- 2. Dept. of Computer Science & Engineering, University of Washington, Seattle, WA
- 3. Department of Computer Science, University of California at San Diego, La Jolla, CA