First-order Fragments with Successor over Infinite Words

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 Σ2 = Σ2[<,+1] and FO2 = FO2[<,+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 FO2 ∩Σ2 if and only if L is the interior of an FO2 language. Symmetrically, a language is in FO2 ∩Π2 if and only if it is the topological closure of an FO2 language. The fragment ∆2 = Σ2∩Π2 contains exactly the clopen languages in FO2. In particular, over infinite words ∆2 is a strict subclass of FO2. 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 FO2 over infinite words was shown 1998 by Wilke in his habilitation thesis whereas decidability of Σ2 over infinite words is new. 1998 ACM Subject Classification F.4.1 Mathematical Logic, F.4.3 Formal Languages.