Continuous separation of game languages

Abstract

We show that a family of tree languages W(';^), previously used by J. Brado/eld, and by the o/rst author to show the strictness of the Mostowski index hierarchy of alternating tree automata, forms a hierarchy w.r.t. the Wadge reducibility. That is, W(';^) ^W W('0;^0) if and only if the index ('0; ^0) is above ('; ^). This is one of the few separation results known so far, concerning the topological complexity of nondeterministically recognizable tree languages, and one of the few results about finite-state recognizable non-Borel sets of trees. The interest of the result is reinforced by the fact that a related family M(';^), witnessing the strictness of the index hierarchy of non-deterministic automata, does not have a similar property