Beyersdorff, O orcid.org/0000-0002-2870-1648 and Pich, J (2016) Understanding Gentzen and Frege Systems for QBF. In: LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science. 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 05-08 Jul 2016, New York, NY, USA. ACM , pp. 146-155. ISBN 978-1-4503-4391-6
Abstract
Recently Beyersdorff, Bonacina, and Chew [10] introduced a natural class of Frege systems for quantified Boolean formulas (QBF) and showed strong lower bounds for restricted versions of these systems. Here we provide a comprehensive analysis of the new extended Frege system from [10], denoted EF + ∀red, which is a natural extension of classical extended Frege EF. Our main results are the following: Firstly, we prove that the standard Gentzen-style system G*1 p-simulates EF + ∀red and that G*1 is strictly stronger under standard complexity-theoretic hardness assumptions.
Secondly, we show a correspondence of EF + ∀red to bounded arithmetic: EF + ∀red can be seen as the non-uniform propositional version of intuitionistic S12. Specifically, intuitionistic S12 proofs of arbitrary statements in prenex form translate to polynomial-size EF + ∀red proofs, and EF + ∀red is in a sense the weakest system with this property. Finally, we show that unconditional lower bounds for EF + ∀red would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold as well. Therefore, the system EF + ∀red naturally unites the central problems from circuit and proof complexity.
Technically, our results rest on a formalised strategy extraction theorem for EF + ∀red akin to witnessing in intuitionistic S12 and a normal form for EF + ∀red proofs.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016 ACM. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in LICS '16 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, http://doi.acm.org/10.1145/2933575.2933597. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | QBF proof systems, sequent calculus, Frege systems, intuitionistic logic, strategy extraction, lower bounds, simulations |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Funding Information: | Funder Grant number EPSRC EP/L024233/1 John Templeton Foundation - DO NOT USE 48138 |
Depositing User: | Symplectic Publications |
Date Deposited: | 11 Apr 2016 15:10 |
Last Modified: | 06 Feb 2020 13:31 |
Published Version: | https://doi.org/10.1145/2933575.2933597 |
Status: | Published |
Publisher: | ACM |
Identification Number: | 10.1145/2933575.2933597 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:98002 |