On the Borel Complexity of MSO Definable Sets of Branches

Abstract

An infinite binaryword can be identified with a branch in the full binary tree. We consider sets of branches definable in monadic second-order logic over the tree, where we allow some extra monadic predicates on the nodes. We show that this class equals to the Boolean combinations of sets in the Borel class Σ $^0_2$ over the Cantor discontinuum. Note that the last coincides with the Borel complexity of ω-regular languages.