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.
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Pitassi, T., Beame, P. & Impagliazzo, R. Exponential lower bounds for the pigeonhole principle. Comput Complexity 3, 97–140 (1993). https://doi.org/10.1007/BF01200117
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DOI: https://doi.org/10.1007/BF01200117