On Recognizable Tree Languages Beyond the Borel Hierarchy

• Authors: Olivier Finkel;Pierre Simonnet
• Venue: Fundamenta Informaticae
• Year: 2009

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 ⩾ 1, there is a D$_{ω^n}$(Σ$^1_1$)-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$_{(ı,κ)}$, for Mostowski-Rabin indices (ıκ). We prove that the D$_{ω^n}$(Σ$^1_1$)-complete tree languages L$_n$ are Wadge reducible to the game tree language W$_{(ı,κ)}$ for κ−ı⩾ 2. In particular these languages W$_{(ı,κ)}$ are not in any class D$_{α}$(Σ$^1_1$) for α