License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2016.51
URN: urn:nbn:de:0030-drops-64637
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6463/
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Itsykson, Dmitry ; Okhotin, Alexander ; Oparin, Vsevolod

Computational and Proof Complexity of Partial String Avoidability

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Abstract

The partial string avoidability problem, also known as partial word avoidability, is stated as follows: given a finite set of strings with possible ``holes'' (undefined symbols), determine whether there exists any two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol.
The problem is known to be NP-hard and in PSPACE, and this paper establishes its PSPACE-completeness.
Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form (CNF) satisfiability problem (SAT), with each clause having infinitely many shifted variants.
Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting constraints (such as clauses, inequalities, polynomials, etc).
Two results on their proof complexity are established.
First, there is a particular formula that has a short refutation in Resolution with shift, but requires classical proofs of exponential size (Resolution, Cutting Plane, Polynomial Calculus, etc.).
At the same time, exponential lower bounds for shifted versions of classical proof systems are established.

BibTeX - Entry

@InProceedings{itsykson_et_al:LIPIcs:2016:6463,
  author =	{Dmitry Itsykson and Alexander Okhotin and Vsevolod Oparin},
  title =	{{Computational and Proof Complexity of Partial String Avoidability}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{51:1--51:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6463},
  URN =		{urn:nbn:de:0030-drops-64637},
  doi =		{10.4230/LIPIcs.MFCS.2016.51},
  annote =	{Keywords: partial strings, partial words, avoidability, proof complexity, PSPACE-completeness}
}

Keywords: partial strings, partial words, avoidability, proof complexity, PSPACE-completeness
Collection: 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)
Issue Date: 2016
Date of publication: 19.08.2016

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