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Parameterized Bounded-Depth Frege Is not Optimal

Published:01 September 2012Publication History
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Abstract

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.

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        • Published in

          cover image ACM Transactions on Computation Theory
          ACM Transactions on Computation Theory  Volume 4, Issue 3
          September 2012
          46 pages
          ISSN:1942-3454
          EISSN:1942-3462
          DOI:10.1145/2355580
          Issue’s Table of Contents

          Copyright © 2012 ACM

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 1 September 2012
          • Accepted: 1 July 2012
          • Revised: 1 March 2012
          • Received: 1 September 2011
          Published in toct Volume 4, Issue 3

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