Regular tree languages in low levels of the Wadge Hierarchy

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).