Abstract.
We consider a modification of the pigeonhole principle, M P H P, introduced by Goerdt in [7]. M P H P is defined over n pigeons and log n holes, and more than one pigeon can go into a hole (according to some rules). Using a technique of Razborov [9] and simplified by Impagliazzo, Pudlák and Sgall [8], we prove that any Polynomial Calculus refutation of a set of polynomials encoding the M P H P, requires degree Ω(log n). We also prove a simple Lemma giving a simulation of Resolution by Polynomial Calculus. Using this lemma, and a Resolution upper bound by Goerdt [7], we obtain that the degree lower bound is tight.
Our lower bound establishes the optimality of the tree-like Resolution simulation by the Polynomial Calculus given in [6].
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Received: 29 March 2001 / Published online: 2 September 2002
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ID="⋆" A prelimianry version appeared as part of the paper A Study of Proof Search Algorithms for Resolution and Polynomial Calculus published in the Proceedings of the 40-th IEEE Conference on Foundations of Computer Science, 1999
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ID="†" Partly supported by Project CICYT, TIC 98-0410-C02-01 and Promoción General del Conocimiento PB98-0937-C04-03.
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ID="††" Part of this work was done while the author was a member of the School of Mathematics at Institute for Advanced Study supported by a NSF grant n. 9987845
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Bonet, M., Galesi, N. Degree complexity for a modified pigeonhole principle. Arch. Math. Logic 42, 403–414 (2003). https://doi.org/10.1007/s001530200141
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DOI: https://doi.org/10.1007/s001530200141