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On the Relative Strength of Pebbling and Resolution
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
2012 (English)In: ACM Transactions on Computational Logic, ISSN 1529-3785, Vol. 13, no 2, 16- p.Article in journal (Refereed) Published
Abstract [en]

The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven to be a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. The typical approach has been to encode the pebble game played on a graph as a CNF formula and then argue that proofs of this formula must inherit (various aspects of) the pebbling properties of the underlying graph. Unfortunately, the reductions used here are not tight. To simulate resolution proofs by pebblings, the full strength of nondeterministic black-white pebbling is needed, whereas resolution is only known to be able to simulate deterministic black pebbling. To obtain strong results, one therefore needs to find specific graph families which either have essentially the same properties for black and black-white pebbling (not at all true in general) or which admit simulations of black-white pebblings in resolution. This article contributes to both these approaches. First, we design a restricted form of black-white pebbling that can be simulated in resolution and show that there are graph families for which such restricted pebblings can be asymptotically better than black pebblings. This proves that, perhaps somewhat unexpectedly, resolution can strictly beat black-only pebbling, and in particular that the space lower bounds on pebbling formulas in Ben-Sasson and Nordstr " om [2008] are tight. Second, we present a versatile parametrized graph family with essentially the same properties for black and black-white pebbling, which gives sharp simultaneous trade-offs for black and black-white pebbling for various parameter settings. Both of our contributions have been instrumental in obtaining the time-space trade-off results for resolution-based proof systems in Ben-Sasson and Nordstr " om [2011].

Place, publisher, year, edition, pages
2012. Vol. 13, no 2, 16- p.
Keyword [en]
Proof complexity, resolution, pebble games, pebbling formula, space, trade-off
National Category
Computer and Information Science
URN: urn:nbn:se:kth:diva-95105DOI: 10.1145/2159531.2159538ISI: 000303047800007ScopusID: 2-s2.0-84860287678OAI: diva2:528097
Swedish Research Council, 621-2010-4797
QC 20120524Available from: 2012-05-24 Created: 2012-05-14 Last updated: 2012-05-24Bibliographically approved

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