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Towards an optimal separation of space and lenght in resolution
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-5379-345X
2013 (English)In: Theory of Computing, ISSN 1557-2862, Vol. 9, 471-557 p.Article in journal (Refereed) Published
Abstract [en]

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of a largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space has been quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether these measures are unrelated in the sense that short proofs can be arbitrarily complex with respect to space.

In this paper, we present some evidence indicating that the latter case should hold and provide a roadmap for how such an optimal separation result could be obtained. We do so by proving a tight bound of Θ(n√) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width of Nordström (STOC 2006) from logarithmic to polynomial.

Place, publisher, year, edition, pages
2013. Vol. 9, 471-557 p.
Keyword [en]
proof complexity, resolution, length, space, width, separation, lower bound, pebbling, black-white pebble game
National Category
Engineering and Technology
URN: urn:nbn:se:kth:diva-138562DOI: 10.4086/toc.2013.v009a014OAI: diva2:681387

QC 20140204

Available from: 2013-12-19 Created: 2013-12-19 Last updated: 2014-02-04Bibliographically approved

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Nordström, JakobHåstad, Johan
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