On Unambiguous Regular Tree Languages of Index (0,2)

Jacques Duparc, Kevin Fournier, Szczepan Hummel
unpublished
Unambiguous automata are usually seen as a natural class of automata in-between deterministic and nondeterministic ones. We show that in case of infinite tree languages, the unambiguous ones are topologically far more complicated than the deterministic ones. We do so by providing operations that generate a family A b α α<ϕ 2 (0) of unambiguous automata such that: 1. It respects the strict Wadge ordering: α < β if and only if A b α < W A b β. This can be established without the help of any
more » ... inacy principle, simply by providing effective winning strategies in the underlying games. 2. Its length (ϕ 2 (0)) is the first fixpoint of the ordinal function that itself enumerates all fixpoints of the ordinal exponentiation x ω x : an ordinal tremendously larger than (ω ω) 3 + 3 which is the height of the Wadge hierarchy of deterministic tree languages as uncovered by Filip Murlak. 3. The priorities of all these parity automata only range from 0 to 2.
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