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Parameterized Bounded-Depth Frege Is not Optimal

Beyersdorff, O, Galesi, N, Lauria, M and Razborov, AA (2012) Parameterized Bounded-Depth Frege Is not Optimal. ACM Transactions on Computation Theory, 4 (3). 7.1 - 7.16 . ISSN 1942-3454

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A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results on tree-like Parameterized Resolution---a parameterized version of classical Resolution---and their gap complexity theorem implies lower bounds for that system. The main result of this article significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size nΩ(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in Dantchev et al. [2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs.

Item Type: Article
Copyright, Publisher and Additional Information: © 2012, Association for Computing Machinery. This is an author produced version of a paper published in ACM Transactions on Computation Theory. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: Algorithms, Theory, Proof complexity, Parameterized complexity, Resolution, bounded-depth Frege
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Engineering (Leeds) > School of Computing (Leeds) > Institute for Computational and Systems Science (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 03 Dec 2012 09:55
Last Modified: 06 Jun 2014 20:57
Published Version: http://dx.doi.org/10.1145/2355580.2355582
Status: Published
Publisher: Association for Computing Machinery
Identification Number: 10.1145/2355580.2355582
URI: http://eprints.whiterose.ac.uk/id/eprint/74772

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