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From small space to small width in resolution
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0003-4003-3168
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
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2015 (English)In: ACM Transactions on Computational Logic, ISSN 1529-3785, E-ISSN 1557-945X, Vol. 16, no 4, 28Article in journal (Refereed) Published
Abstract [en]

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexitymeasure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similarmethods.

Place, publisher, year, edition, pages
2015. Vol. 16, no 4, 28
Keyword [en]
PCR, Polynomial calculus, Proof complexity, Resolution, Space, Width
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-174699DOI: 10.1145/2746339ISI: 000365216500001Scopus ID: 2-s2.0-84941557324OAI: oai:DiVA.org:kth-174699DiVA: diva2:885104
Funder
EU, FP7, Seventh Framework ProgrammeEU, European Research Council, 621-2010-4797, 621-2012-5645Swedish Research Council, 621-2010-4797, 621-2012-5645
Note

QC 20151111

Available from: 2015-11-11 Created: 2015-10-07 Last updated: 2017-12-01Bibliographically approved
In thesis
1. On Complexity Measures in Polynomial Calculus
Open this publication in new window or tab >>On Complexity Measures in Polynomial Calculus
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Proof complexity is the study of different resources that a proof needs in different proof systems for propositional logic. This line of inquiry relates to the fundamental questions in theoretical computer science, as lower bounds on proof size for an arbitrary proof system would separate P from NP.

We study two simple proof systems: resolution and polynomial calculus. In resolution we reason using clauses, while in polynomial calculus we use polynomials. We study three measures of complexity of proofs: size, space, and width/degree. Size is the number of clauses or monomials that appear in a resolution or polynomial calculus proof, respectively. Space is the maximum number of clauses/monomials we need to keep at each time step of the proof. Width/degree is the size of the largest clause/monomial in a proof.

Width is a lower bound for space in resolution. The original proof of this claim used finite model theory. In this thesis we give a different, more direct proof of the space-width relation. We can ask whether a similar relation holds between space and degree in polynomial calculus. We make some progress on this front by showing that when a formula F requires resolution width w then the XORified version of F requires polynomial calculus space Ω(w). We also show that space lower bounds do not imply degree lower bounds in polynomial calculus.

Width/degree and size are also related, as strong lower bounds for width/degree imply strong lower bounds for size. Currently, proving width lower bounds has a well-developed machinery behind it. However, the degree measure is much less well-understood. We provide a unified framework for almost all previous degree lower bounds. We also prove some new degree and size lower bounds. In addition, we explore the relation between theory and practice by running experiments on some current state-of-the-art SAT solvers.

Place, publisher, year, edition, pages
Stockholm, Sweden: KTH Royal Institute of Technology, 2016. 180 p.
Series
TRITA-CSC-A, ISSN 1653-5723 ; 2017:02
National Category
Computer Science
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-197278 (URN)978-91-7729-226-5 (ISBN)
Public defence
2017-01-20, D2, Lindstedtsvägen 5, Stockholm, 14:00 (English)
Opponent
Supervisors
Projects
Understanding the Hardness of Theorem Proving
Funder
EU, FP7, Seventh Framework Programme, 279611
Note

QC 20161206

Available from: 2016-12-06 Created: 2016-11-30 Last updated: 2016-12-26Bibliographically approved

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Lauria, MassimoNordström, Jakob

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