We show that a family of tree languages W$_{(l,k)}$, previously used by J. Bradfield, and by the first author to show the strictness of the Rabin¨CMostowski index hierarchy of alternating tree automata, forms a hierarchy w.r.t. theWadge reducibility. That is, W$_{(l,k)$ l$_W$ W$_{(l',k')}$ if and only if the index (l', k') is above (l, k). This is one of the few separation results known so far, concerning the topological complexity of non-deterministically recognizable tree languages, and one of the few results about finite-state recognizable non-Borel sets of trees.
['André Arnold', 'Damian Niwiński']
Continuous Separation of Game Languages