An Upper Bound on the Complexity of Recognizable Tree Languages

TL;DR

This paper explores the topological complexity of regular tree languages, establishing an upper bound within a specific class and demonstrating it is significantly better than the general ${\bf\Delta}^1_2$ bound.

Contribution

The paper provides a detailed exposition of the upper bound on the complexity of regular tree languages and relates it to the Wadge hierarchy and Veblen functions.

Findings

Regular tree languages are in the class $\Game (D_n({\bf\Sigma}^0_2))$ for some $n\geq 1$

The upper bound on their complexity is much better than ${\bf\Delta}^1_2$

Embedding of the Wadge hierarchy shows a refined topological complexity bound.

Abstract

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bf\Sigma}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ 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 $2^\omega$ into the class ${\bf\Delta}^1\_2$, 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 ${\bf\Delta}^1\_2$.