Infinite trees and automaton- definable relations over ω-words

Abstract

We study relations over ω-words using a representation by tree languages. An ω-word over an alphabet with k letters is considered as a path through the k-ary tree, an n-tuple of ω-words as an n-tuple of paths (coded by an appropriate valuation of the k-ary tree using values in {0,1} n ), and a relation over ω-words as a tree language. In the first part of the paper we give a logical characterization of the “Rabin-recognizable relations” (whose associated tree languages are recognized by Rabin tree automata) in terms of “weak chain logic”, a restriction of monadic second-order logic over trees. In the second part of the paper an extended logic is considered, obtained by adjoining the “equal-level predicate” over trees. We describe the class of relations over ω-words which (in the tree language representation) are definable in this logic, and show that the theory of the k-ary tree in this logic is decidable. It covers tree properties which are not expressible in the monadic second-order logic S kS.