The Non-deterministic Mostowski Hierarchy and Distance-Parity Automata

Given a Rabin tree-language and natural numbers i, j, the language is said to be i, j-feasible if it is accepted by a parity automaton using priorities {i, i+1, ..., j}. The i, j-feasibility induces a hierarchy over the Rabin-tree languages called the Mostowski hierarchy. In this paper we prove that the problem of deciding if a language is i, j-feasible is reducible to the uniform universality problem for distanceparity automata. Distance-parity automata form a new model of automata extending both the nested distance desert automata introduced by Kirsten in his proof of decidability of the star-height problem, and parity automata over infinite trees. Distance-parity automata, instead of accepting a language, attach to each tree a cost in ω + 1. The uniform universality problem consists in determining if this cost function is bounded by a finite value.