First-order fragments with successor over infinite words

Abstract

We consider fragments of first-order logic and as models we allow finite andinfinite words simultaneously. The only binary relations apart from equalityare order comparison < and the successor predicate +1. We givecharacterizations of the fragments Sigma2 = Sigma2[<,+1] and FO2 = FO2[<,+1] interms of algebraic and topological properties. To this end we introduce thefactor topology over infinite words. It turns out that a language L is in theintersection of FO2 and Sigma2 if and only if L is the interior of an FO2language. Symmetrically, a language is in the intersection of FO2 and Pi2 ifand only if it is the topological closure of an FO2 language. The fragmentDelta2, which by definition is the intersection Sigma2 and Pi2 contains exactlythe clopen languages in FO2. In particular, over infinite words Delta2 is astrict subclass of FO2. Our characterizations yield decidability of themembership problem for all these fragments over finite and infinite words; andas a corollary we also obtain decidability for infinite words. Moreover, wegive a new decidable algebraic characterization of dot-depth 3/2 over finitewords. Decidability of dot-depth 3/2 over finite words was first shown by Glaßer andSchmitz in STACS 2000, and decidability of the membership problem for FO2 overinfinite words was shown 1998 by Wilke in his habilitation thesis whereasdecidability of Sigma2 over infinite words was not known before.