Itsykson, Dmitry ;
Okhotin, Alexander ;
Oparin, Vsevolod
Computational and Proof Complexity of Partial String Avoidability
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 twosided infinite string containing no substrings from this set, assuming that a hole matches every symbol.
The problem is known to be NPhard and in PSPACE, and this paper establishes its PSPACEcompleteness.
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.
Nonsatisfiability 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 = {51:151:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770163},
ISSN = {18688969},
year = {2016},
volume = {58},
editor = {Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6463},
URN = {urn:nbn:de:0030drops64637},
doi = {10.4230/LIPIcs.MFCS.2016.51},
annote = {Keywords: partial strings, partial words, avoidability, proof complexity, PSPACEcompleteness}
}
2016
Keywords: 

partial strings, partial words, avoidability, proof complexity, PSPACEcompleteness 
Seminar: 

41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Issue date: 

2016 
Date of publication: 

2016 