eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2008-02-06
385
396
10.4230/LIPIcs.STACS.2008.1360
article
Cardinality and counting quantifiers on omega-automatic structures
Kaiser, Lukasz
Rubin, Sasha
Bárány, Vince
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol001-stacs2008/LIPIcs.STACS.2008.1360/LIPIcs.STACS.2008.1360.pdf
$omega$-automatic presentations
$omega$-semigroups
$omega$-automata