An upper bound on the complexity of recognizable tree languages

Abstract

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class (D n (Σ 0 2 )) for some natural number n ≥ 1, where is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space 2 ω into the class ∆ 1 2 , and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ∆ 1 2 .