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On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
2012 (English)In: Proceedings of the Annual ACM Symposium on Theory of Computing, 2012, 233-247 p.Conference paper (Refereed)
Abstract [en]

An active line of research in proof complexity over the last decade has been the study of proof space and trade-offs between size and space. Such questions were originally motivated by practical SAT solving, but have also led to the development of new theoretical concepts in proof complexity of intrinsic interest and to results establishing nontrivial relations between space and other proof complexity measures. By now, the resolution proof system is fairly well understood in this regard, as witnessed by a sequence of papers leading up to [Ben-Sasson and Nordstrom 2008, 2011] and [Beame, Beck, and Impagliazzo 2012]. However, for other relevant proof systems in the context of SAT solving, such as polynomial calculus (PC) and cutting planes (CP), very little has been known. Inspired by [BN08, BN11], we consider CNF encodings of so-called pebble games played on graphs and the approach of making such pebbling formulas harder by simple syntactic modifications. We use this paradigm of hardness amplification to make progress on the relatively longstanding open question of proving time-space trade-offs for PC and CP. Namely, we exhibit a family of modified pebbling formulas {F n} n such that: • The formulas F n have size Θ(n) and width O(1). • They have proofs in length O(n) in resolution, which generalize to both PC and CP. • Any refutation in CP or PCR (a generalization of PC) in length L and space s must satisfy s log L > 4√n. A crucial technical ingredient in these results is a new two-player communication complexity lower bound for composed search problems in terms of block sensitivity, a contribution that we believe to be of independent interest.

Place, publisher, year, edition, pages
2012. 233-247 p.
Keyword [en]
cutting planes, length, pcr, polynomial calculus, proof complexity, resolution, size, space, trade-offs
National Category
Computer Science
URN: urn:nbn:se:kth:diva-112875DOI: 10.1145/2213977.2214000ScopusID: 2-s2.0-84862626028OAI: diva2:587537
44th Annual ACM Symposium on Theory of Computing, STOC '12; New York, NY; United States

QC 20130523

Available from: 2013-01-14 Created: 2013-01-14 Last updated: 2013-09-16Bibliographically approved

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