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URN: urn:nbn:de:0030-drops-9751
URL: http://drops.dagstuhl.de/opus/volltexte/2007/975/

Weis, Philipp ; Immerman, Neil

Structure Theorem and Strict Alternation Hierarchy for FO² on Words

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Abstract

It is well-known that every first-order property on words is expressible using at most three variables. The subclass of properties expressible with only two variables is also quite interesting and well-studied. We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words. Our results apply to FO$^2[<]$ and FO$^2[<,suc]$, the latter of which includes the binary successor relation in addition to the linear ordering on string positions. For both languages, our structure theorems show exactly what is expressible using a given quantifier depth, $n$, and using $m$ blocks of alternating quantifiers, for any $mleq n$. Using these characterizations, we prove, among other results, that there is a strict hierarchy of alternating quantifiers for both languages. The question whether there was such a hierarchy had been completely open since it was asked in [Etessami, Vardi, and Wilke 1997].

BibTeX - Entry

@InProceedings{weis_et_al:DSP:2007:975,
  author =	{Philipp Weis and Neil Immerman},
  title =	{Structure Theorem and Strict Alternation Hierarchy for FO² on Words},
  booktitle =	{Circuits, Logic, and Games},
  year =	{2007},
  editor =	{Thomas Schwentick and Denis Th{\'e}rien and Heribert Vollmer },
  number =	{06451},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2007/975},
  annote =	{Keywords: Descriptive complexity, finite model theory, alternation hierarchy,  Ehrenfeucht-Fraisse games}
}

Keywords: Descriptive complexity, finite model theory, alternation hierarchy, Ehrenfeucht-Fraisse games
Seminar: 06451 - Circuits, Logic, and Games
Issue date: 2007
Date of publication: 23.04.2007


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