Fragments of First-Order Logic over Infinite Words

Abstract

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO[<] for ω-languages: Σ2, FO2, FO2∩Σ2, and Δ2 (and by duality Π2 and FO2∩Π2). 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 (Classifying Discrete Temporal Properties. Habilitationsschrift, Universitat Kiel, April 1998) and Bojanczyk (Lecture Notes in Computer Science, vol. 4962, pp. 172–185, 2008) and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.