10.4230/LIPICS.STACS.2011.356
Kallas, Jakub
Jakub
Kallas
Kufleitner, Manfred
Manfred
Kufleitner
Lauser, Alexander
Alexander
Lauser
First-order Fragments with Successor over Infinite Words
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2011
Article
infinite words
regular languages
first-order logic
automata theory
semi-groups
topology
Schwentick, Thomas
Thomas
Schwentick
Dürr, Christoph
Christoph
Dürr
2011
2011-03-11
2011-03-11
2011-03-11
en
urn:nbn:de:0030-drops-30267
10.4230/LIPIcs.STACS.2011
978-3-939897-25-5
1868-8969
10.4230/LIPIcs.STACS.2011
LIPIcs, Volume 9, STACS 2011
28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
2013
9
30
356
367
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Schwentick, Thomas
Thomas
Schwentick
Dürr, Christoph
Christoph
Dürr
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2011
9
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
736860 bytes
application/pdf
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
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.
LIPIcs, Vol. 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011), pages 356-367