article

Hard examples for the bounded depth Frege proof system

Author:

Eli Ben-SassonAuthors Info & Claims

Computational Complexity, Volume 11, Issue 3/4

Pages 109 - 136

https://doi.org/10.1007/s00037-002-0172-5

Published: 01 March 2003 Publication History

Abstract

We prove exponential lower bounds on the size of a bounded depth Frege proof of a Tseitin graph-based contradiction, whenever the underlying graph is an expander. This is the first example of a contradiction, naturally formalized as a 3-CNF, that has no short bounded depth Frege proofs. Previously, lower bounds of this type were known only for the pigeonhole principle and for Tseitin contradictions based on complete graphs.Our proof is a novel reduction of a Tseitin formula of an expander graph to the pigeonhole principle, in a manner resembling that done by Fu and Urquhart for complete graphs.In the proof we introduce a general method for removing extension variables without significantly increasing the proof size, which may be interesting in its own right.

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Cited By

Vinall-Smeeth HSobocinski PLago UEsparza J(2024)From Quantifier Depth to Quantifier Number: Separating Structures with k VariablesProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662125(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662125
Berkholz CNordström J(2023)Near-optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement StepsJournal of the ACM10.1145/319525770:5(1-32)Online publication date: 11-Oct-2023
https://dl.acm.org/doi/10.1145/3195257
Håstad J(2020)On Small-depth Frege Proofs for Tseitin for GridsJournal of the ACM10.1145/342560668:1(1-31)Online publication date: 17-Nov-2020
https://dl.acm.org/doi/10.1145/3425606
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Hard examples for the bounded depth Frege proof system

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cover image Computational Complexity

Computational Complexity Volume 11, Issue 3/4

March 2003

80 pages

ISSN:1016-3328

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Birkhauser Verlag

Switzerland

Publication History

Published: 01 March 2003

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Cited By

Vinall-Smeeth HSobocinski PLago UEsparza J(2024)From Quantifier Depth to Quantifier Number: Separating Structures with k VariablesProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662125(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662125
Berkholz CNordström J(2023)Near-optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement StepsJournal of the ACM10.1145/319525770:5(1-32)Online publication date: 11-Oct-2023
https://dl.acm.org/doi/10.1145/3195257
Håstad J(2020)On Small-depth Frege Proofs for Tseitin for GridsJournal of the ACM10.1145/342560668:1(1-31)Online publication date: 17-Nov-2020
https://dl.acm.org/doi/10.1145/3425606
Gryaznov S(2019)Notes on Resolution over Linear EquationsComputer Science – Theory and Applications10.1007/978-3-030-19955-5_15(168-179)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1007/978-3-030-19955-5_15
Berkholz CNordström JKoskinen EGrohe MShankar N(2016)Near-Optimal Lower Bounds on Quantifier Depth and Weisfeiler--Leman Refinement StepsProceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/2933575.2934560(267-276)Online publication date: 5-Jul-2016
https://dl.acm.org/doi/10.1145/2933575.2934560
Atserias A(2013)The proof-search problem between bounded-width resolution and bounded-degree semi-algebraic proofsProceedings of the 16th international conference on Theory and Applications of Satisfiability Testing10.1007/978-3-642-39071-5_1(1-17)Online publication date: 8-Jul-2013
https://dl.acm.org/doi/10.1007/978-3-642-39071-5_1
Impagliazzo RSegerlind N(2006)Constant-depth Frege systems with counting axioms polynomially simulate Nullstellensatz refutationsACM Transactions on Computational Logic (TOCL)10.1145/1131313.11313147:2(199-218)Online publication date: 1-Apr-2006
https://dl.acm.org/doi/10.1145/1131313.1131314

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