Fragments of First-Order Logic over Infinite Words
We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic $\mathrm{FO}[<]$ for $\omega$-languages: $\Sigma_2$, $\Delta_2$, $\mathrm{FO}^2 \cap \Sigma_2$ (and by duality $\mathrm{FO}^2 \cap \Pi_2$), and $\mathrm{FO}^2$. These descriptions extend the respective results for finite words. In particular, we relate the above fragments to language classes of certain (unambiguous) polynomials. An immediate consequence is the decidability of the membership problem of these classes, but this was shown before by Wilke (1998) and Boja{\'n}czyk (2008) and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.
Infinite words
Regular languages
First-order logic
Automata theory
Semigroups
Topology
325-336
Regular Paper
Volker
Diekert
Volker Diekert
Manfred
Kufleitner
Manfred Kufleitner
10.4230/LIPIcs.STACS.2009.1818
Creative Commons Attribution-NoDerivs 3.0 Unported license
https://creativecommons.org/licenses/by-nd/3.0/legalcode