License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2008.1360
URN: urn:nbn:de:0030-drops-13602
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2008/1360/
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Kaiser, Lukasz ; Rubin, Sasha ; Bárány, Vince

Cardinality and counting quantifiers on omega-automatic structures

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Abstract

We investigate structures that can be represented by
omega-automata, so called omega-automatic structures, and prove
that relations defined over such structures in first-order logic
expanded by the first-order quantifiers `there exist at most
$aleph_0$ many', 'there exist finitely many' and 'there exist $k$
modulo $m$ many' are omega-regular. The proof identifies certain
algebraic properties of omega-semigroups.

As a consequence an omega-regular equivalence relation of countable
index has an omega-regular set of representatives. This implies
Blumensath's conjecture that a countable structure with an
$omega$-automatic presentation can be represented using automata
on finite words. This also complements a very recent result of
Hj"orth, Khoussainov, Montalban and Nies showing that there is an
omega-automatic structure which has no injective presentation.


BibTeX - Entry

@InProceedings{kaiser_et_al:LIPIcs:2008:1360,
  author =	{Lukasz Kaiser and Sasha Rubin and Vince B{\'a}r{\'a}ny},
  title =	{{Cardinality and counting quantifiers on omega-automatic structures}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{385--396},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1360},
  URN =		{urn:nbn:de:0030-drops-13602},
  doi =		{10.4230/LIPIcs.STACS.2008.1360},
  annote =	{Keywords: $omega$-automatic presentations, $omega$-semigroups, $omega$-automata}
}

Keywords: $omega$-automatic presentations, $omega$-semigroups, $omega$-automata
Collection: 25th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2008
Date of publication: 06.02.2008

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