research-article

Parameterized Bounded-Depth Frege Is not Optimal

Authors:

Olaf Beyersdorff,

Massimo Lauria,

Alexander A. RazborovAuthors Info & Claims

ACM Transactions on Computation Theory (TOCT), Volume 4, Issue 3

Article No.: 7, Pages 1 - 16

https://doi.org/10.1145/2355580.2355582

Published: 01 September 2012 Publication History

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 n^Ω(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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Cited By

Dantchev SGalesi NGhani AMartin B(2024)Proof Complexity and the Binary Encoding of Combinatorial PrinciplesSIAM Journal on Computing10.1137/20M134784X53:3(764-802)Online publication date: 24-Jun-2024
https://doi.org/10.1137/20M134784X
De Rezende SPotechin ARisse K(2023)Clique Is Hard on Average for Unary Sherali-Adams2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00008(12-25)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00008
Atserias ABonacina IDe Rezende SLauria MNordström JRazborov A(2021)Clique Is Hard on Average for Regular ResolutionJournal of the ACM10.1145/344935268:4(1-26)Online publication date: 30-Jun-2021
https://dl.acm.org/doi/10.1145/3449352
Show More Cited By

Index Terms

Parameterized Bounded-Depth Frege Is not Optimal
1. Theory of computation
  1. Computational complexity and cryptography
    1. Interactive proof systems
    2. Proof complexity
  2. Logic
    1. Proof theory

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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.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 September 2012

Accepted: 01 July 2012

Revised: 01 March 2012

Received: 01 September 2011

Published in TOCT Volume 4, Issue 3

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Cited By

Dantchev SGalesi NGhani AMartin B(2024)Proof Complexity and the Binary Encoding of Combinatorial PrinciplesSIAM Journal on Computing10.1137/20M134784X53:3(764-802)Online publication date: 24-Jun-2024
https://doi.org/10.1137/20M134784X
De Rezende SPotechin ARisse K(2023)Clique Is Hard on Average for Unary Sherali-Adams2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00008(12-25)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00008
Atserias ABonacina IDe Rezende SLauria MNordström JRazborov A(2021)Clique Is Hard on Average for Regular ResolutionJournal of the ACM10.1145/344935268:4(1-26)Online publication date: 30-Jun-2021
https://dl.acm.org/doi/10.1145/3449352
Sigley SBeyersdorff O(2021)Proof Complexity of Modal ResolutionJournal of Automated Reasoning10.1007/s10817-021-09609-9Online publication date: 13-Oct-2021
https://doi.org/10.1007/s10817-021-09609-9
Dantchev SGalesi NMartin BAceto L(2019)Resolution and the binary encoding of combinatorial principlesProceedings of the 34th Computational Complexity Conference10.4230/LIPIcs.CCC.2019.6(1-25)Online publication date: 17-Jul-2019
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2019.6
Atserias ABonacina Ide Rezende SLauria MNordström JRazborov ADiakonikolas IKempe DHenzinger M(2018)Clique is hard on average for regular resolutionProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188856(866-877)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188856
Lauria M(2018)Cliques enumeration and tree-like resolution proofsInformation Processing Letters10.1016/j.ipl.2018.03.001135(62-67)Online publication date: Jul-2018
https://doi.org/10.1016/j.ipl.2018.03.001
Lauria MPudlák PRödl VThapen N(2017)The complexity of proving that a graph is RamseyCombinatorica10.1007/s00493-015-3193-937:2(253-268)Online publication date: 1-Apr-2017
https://dl.acm.org/doi/10.1007/s00493-015-3193-9
Lauria MNordström J(2017)Tight Size-Degree Bounds for Sums-of-Squares ProofsComputational Complexity10.1007/s00037-017-0152-426:4(911-948)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1007/s00037-017-0152-4
Lauria MElffers JNordström JVinyals M(2017)CNFgen: A Generator of Crafted BenchmarksTheory and Applications of Satisfiability Testing – SAT 201710.1007/978-3-319-66263-3_30(464-473)Online publication date: 9-Aug-2017
https://doi.org/10.1007/978-3-319-66263-3_30
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