Fragments of first-order logic over infinite words

TL;DR

This paper characterizes various fragments of first-order logic over infinite words using topological, algebraic, and language-theoretic methods, extending known results from finite words and establishing decidability of class membership.

Contribution

It provides new topological and algebraic characterizations of FO[<] fragments over omega-languages, connecting them to polynomial language classes and extending finite word results.

Findings

Characterizations of FO^2, Sigma_2, Delta_2 over omega-languages

Extension of finite word results to infinite words

Decidability of class membership for these fragments

Abstract

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO[<] for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_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 and Bojanczyk and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.