Exponential lower bounds for the pigeonhole principle
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(loglog
n)-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, log n-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 systems lower bounds Subject classifications 68Q99 03F20 68R05 References
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