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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2011.356
URN: urn:nbn:de:0030-drops-30267
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2011/3026/
Kallas, Jakub ;
Kufleitner, Manfred ;
Lauser, Alexander
First-order Fragments with Successor over Infinite Words
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 = {10.4230/LIPIcs.STACS.2011.356},
annote = {Keywords: infinite words, regular languages, first-order logic, automata theory, semi-groups, topology}
}
Keywords: |
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infinite words, regular languages, first-order logic, automata theory, semi-groups, topology |
Collection: |
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28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) |
Issue Date: |
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2011 |
Date of publication: |
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11.03.2011 |