On Recognizable Tree Languages Beyond the Borel Hierarchy

Overview & Synopsis

We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer $n \geq 1$, there is a $D_{\omega^n}({\bf \Sigma}^1_1)$-complete tree language L_n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Büchi tree automaton must be Borel. Then we consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i, k)$. We prove that the $D_{\omega^n}({\bf \Sigma}^1_1)$-complete tree languages L_n are Wadge reducible to the game tree language $W_{(i, k)}$ for $k-i \geq 2$. In particular these languages $W_{(i, k)}$ are not in any class $D_{\alpha}({\bf \Sigma}^1_1)$ for $\alpha < \omega^\omega$.