First-order Fragments with Successor over Infinite Words
Résumé
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 \lt and the successor predicate $+1$. We give characterizations of the fragments $\Sigma_2 = \Sigma_2[\lt,\suc]$ and $\FO^2 = \FO^2[\lt,\suc]$ 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 $\FO^2 \cap \Sigma_2$ if and only if $L$ is the interior of an $\FO^2$ language. Symmetrically, a language is in $\FO^2 \cap \Pi_2$ if and only if it is the topological closure of an $\FO^2$ language. The fragment $\Delta_2 = \Sigma_2 \cap \Pi_2$ contains exactly the clopen languages in $\FO^2$. In particular, over infinite words $\Delta_2$ is a strict subclass of $\FO^2$. 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 Gla{ß}er and Schmitz in STACS 2000, and decidability of the membership problem for $\FO^2$ over infinite words was shown 1998 by Wilke in his habilitation thesis whereas decidability of $\Sigma_2$ over infinite words is new.
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