computational complexity

, Volume 3, Issue 2, pp 97–140

Exponential lower bounds for the pigeonhole principle

  • Toniann Pitassi
  • Paul Beame
  • Russell Impagliazzo
Article

DOI: 10.1007/BF01200117

Cite this article as:
Pitassi, T., Beame, P. & Impagliazzo, R. Comput Complexity (1993) 3: 97. doi:10.1007/BF01200117

Abstract

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.

Key words

Complexity of propositional proof systemslower bounds

Subject classifications

68Q9903F2068R05

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Toniann Pitassi
    • 1
  • Paul Beame
    • 2
  • Russell Impagliazzo
    • 3
  1. 1.Department of Computer ScienceUniversity of California at San DiegoLa Jolla
  2. 2.Dept. of Computer Science & EngineeringUniversity of WashingtonSeattle
  3. 3.Department of Computer ScienceUniversity of California at San DiegoLa Jolla