Regular tree languages in low levels of the Wadge Hierarchy

TL;DR

This paper characterizes regular infinite tree languages within low Wadge hierarchy levels, providing decidability results for specific classes including Boolean combinations of open sets and the Borel class Delta2.

Contribution

It offers effective characterizations and decidability results for regular tree languages in low Wadge hierarchy levels, including finite levels, BC(Sigma_1^0), and Delta_2^0.

Findings

Decidability for finite levels of the hierarchy

Decidability for BC(Sigma_1^0) class

Decidability for Delta_2^0 Borel class

Abstract

In this article we provide effective characterisations of regular languages of infinite trees that belong to the low levels of the Wadge hierarchy. More precisely we prove decidability for each of the finite levels of the hierarchy; for the class of the Boolean combinations of open sets $BC(\Sigma_1^0)$ (i.e. the union of the first $\omega$ levels); and for the Borel class $\Delta_2^0$ (i.e. for the union of the first $\omega_1$ levels).