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First-order Fragments with Successor over Infinite Words

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Abstract

We consider fragments of first-order logic and as models we allow finite and infinite words simultaneously. The only binary relations apart from equality are order comparison < and the successor predicate +1. We give characterizations of the fragments Sigma_2 = Sigma_2[<,+1] and FO^2 = FO^2[<,+1] in terms of algebraic and topological properties. To this end we introduce the factor topology over infinite words. It turns out that a language $L$ is in FO^2 cap Sigma_2 if and only if $L$ is the interior of an FO^2 language. Symmetrically, a language is in FO^2 cap Pi_2 if and only if it is the topological closure of an FO^2 language. The fragment Delta_2 = Sigma_2 cap Pi_2 contains exactly the clopen languages in FO^2. In particular, over infinite words Delta_2 is a strict subclass of FO^2. Our characterizations yield decidability of the membership problem for all these fragments over finite and infinite words; and as a corollary we also obtain decidability for infinite words. Moreover, we give a new decidable algebraic characterization of dot-depth 3/2 over finite words. Decidability of dot-depth 3/2 over finite words was first shown by Glasser and Schmitz in STACS 2000, and decidability of the membership problem for FO^2 over infinite words was shown 1998 by Wilke in his habilitation thesis whereas decidability of Sigma_2 over infinite words is new.

BibTeX - Entry

@InProceedings{kallas_et_al:LIPIcs:2011:3026,
  author =	{Jakub Kallas and Manfred Kufleitner and Alexander Lauser},
  title =	{{First-order Fragments with Successor over Infinite Words}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
  pages =	{356--367},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Thomas Schwentick and Christoph D{\"u}rr},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2011/3026},
  URN =		{urn:nbn:de:0030-drops-30267},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2011.356},
  annote =	{Keywords: infinite words, regular languages, first-order logic, automata theory, semi-groups, topology}
}

Keywords: infinite words, regular languages, first-order logic, automata theory, semi-groups, topology
Seminar: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Issue date: 2011
Date of publication: 2011


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