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DOI: 10.4230/LIPIcs.ITCS.2018.9
URN: urn:nbn:de:0030-drops-83228
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8322/
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Beyersdorff, Olaf ; Blinkhorn, Joshua ; Hinde, Luke

Size, Cost and Capacity: A Semantic Technique for Hard Random QBFs

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Abstract

As a natural extension of the SAT problem, an array of proof systems for quantified Boolean formulas (QBF) have been proposed, many of which extend a propositional proof system to handle universal quantification. By formalising the construction of the QBF proof system obtained from a propositional proof system by adding universal reduction (Beyersdorff, Bonacina & Chew, ITCS'16), we present a new technique for proving proof-size lower bounds in these systems. The technique relies only on two semantic measures: the cost of a QBF, and the capacity of a proof. By examining the capacity of proofs in several QBF systems, we are able to use the technique to obtain lower bounds based on cost alone. As applications of the technique, we first prove exponential lower bounds for a new family of simple QBFs representing equality. The main application is in proving exponential lower bounds with high probability for a class of randomly generated QBFs, the first 'genuine' lower bounds of this kind, which apply to the QBF analogues of resolution, Cutting Planes, and Polynomial Calculus. Finally, we employ the technique to give a simple proof of hardness for a prominent family of QBFs.

BibTeX - Entry

@InProceedings{beyersdorff_et_al:LIPIcs:2018:8322,
  author =	{Olaf Beyersdorff and Joshua Blinkhorn and Luke Hinde},
  title =	{{Size, Cost and Capacity: A Semantic Technique for Hard Random QBFs}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Anna R. Karlin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8322},
  URN =		{urn:nbn:de:0030-drops-83228},
  doi =		{10.4230/LIPIcs.ITCS.2018.9},
  annote =	{Keywords: quantified Boolean formulas, proof complexity, lower bounds}
}

Keywords: quantified Boolean formulas, proof complexity, lower bounds
Seminar: 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)
Issue Date: 2018
Date of publication: 05.01.2018


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