Abstract
We consider fragments of firstorder 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 dotdepth 3/2 over finite words.
Decidability of dotdepth 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 = {{Firstorder Fragments with Successor over Infinite Words}},
booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
pages = {356367},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897255},
ISSN = {18688969},
year = {2011},
volume = {9},
editor = {Thomas Schwentick and Christoph D{\"u}rr},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3026},
URN = {urn:nbn:de:0030drops30267},
doi = {10.4230/LIPIcs.STACS.2011.356},
annote = {Keywords: infinite words, regular languages, firstorder logic, automata theory, semigroups, topology}
}
Keywords: 

infinite words, regular languages, firstorder logic, automata theory, semigroups, topology 
Collection: 

28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) 
Issue Date: 

2011 
Date of publication: 

11.03.2011 