An upper bound on the complexity of recognizable tree languages

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class a(Dn(Σ2)) for some natural number n ≥ 1, where a 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 into the class ∆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. keywords: infinite trees; tree automaton; regular tree language; Cantor topology; topological complexity; Borel hierarchy; game quantifier; Wadge classes; Wadge degrees; universal sets; provably-∆2.