research-article

Frege Systems for Quantified Boolean Logic

Authors:

Olaf Beyersdorff,

Ilario Bonacina,

Jan PichAuthors Info & Claims

Journal of the ACM (JACM), Volume 67, Issue 2

Article No.: 9, Pages 1 - 36

https://doi.org/10.1145/3381881

Published: 05 April 2020 Publication History

Abstract

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work.

This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC⁰[p] circuits. In contrast, any non-trivial lower bound for propositional AC⁰[p]-Frege constitutes a major open problem.

Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits).

We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.

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Cited By

Beyersdorff OClymo JDantchev SMartin B(2024)The Riis Complexity Gap for QBF ResolutionJournal on Satisfiability, Boolean Modeling and Computation10.3233/SAT-23150515:1(9-25)Online publication date: 29-Oct-2024
https://doi.org/10.3233/SAT-231505
Beyersdorff OBlinkhorn JMahajan MPeitl TSood G(2024)Hard QBFs for Merge ResolutionACM Transactions on Computation Theory10.1145/363826316:2(1-24)Online publication date: 14-Mar-2024
https://dl.acm.org/doi/10.1145/3638263
Böhm BBeyersdorff O(2023)Lower Bounds for QCDCL via Formula GaugeJournal of Automated Reasoning10.1007/s10817-023-09683-167:4Online publication date: 27-Sep-2023
https://doi.org/10.1007/s10817-023-09683-1
Show More Cited By

Index Terms

Frege Systems for Quantified Boolean Logic
1. Theory of computation
  1. Computational complexity and cryptography

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 67, Issue 2

April 2020

123 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3393620

Editor:
Éva Tardos
Cornell University

Issue’s Table of Contents

Copyright © 2020 Owner/Author.

This work is licensed under a Creative Commons Attribution International 4.0 License.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 05 April 2020

Accepted: 01 February 2020

Revised: 01 August 2019

Received: 01 September 2016

Published in JACM Volume 67, Issue 2

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European Research Council under the European Union's Seventh Framework Programme(FP7/2007-2014)/ERC
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Cited By

Beyersdorff OClymo JDantchev SMartin B(2024)The Riis Complexity Gap for QBF ResolutionJournal on Satisfiability, Boolean Modeling and Computation10.3233/SAT-23150515:1(9-25)Online publication date: 29-Oct-2024
https://doi.org/10.3233/SAT-231505
Beyersdorff OBlinkhorn JMahajan MPeitl TSood G(2024)Hard QBFs for Merge ResolutionACM Transactions on Computation Theory10.1145/363826316:2(1-24)Online publication date: 14-Mar-2024
https://dl.acm.org/doi/10.1145/3638263
Böhm BBeyersdorff O(2023)Lower Bounds for QCDCL via Formula GaugeJournal of Automated Reasoning10.1007/s10817-023-09683-167:4Online publication date: 27-Sep-2023
https://doi.org/10.1007/s10817-023-09683-1
Beyersdorff OBlinkhorn J(2021)A simple proof of QBF hardnessInformation Processing Letters10.1016/j.ipl.2021.106093168(106093)Online publication date: Jun-2021
https://doi.org/10.1016/j.ipl.2021.106093
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
Chew L(2021)Hardness and Optimality in QBF Proof Systems Modulo NPTheory and Applications of Satisfiability Testing – SAT 202110.1007/978-3-030-80223-3_8(98-115)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80223-3_8
Böhm BBeyersdorff O(2021)Lower Bounds for QCDCL via Formula GaugeTheory and Applications of Satisfiability Testing – SAT 202110.1007/978-3-030-80223-3_5(47-63)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80223-3_5
Mengel SSlivovsky F(2021)Proof Complexity of Symbolic QBF ReasoningTheory and Applications of Satisfiability Testing – SAT 202110.1007/978-3-030-80223-3_28(399-416)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80223-3_28

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