The Non-deterministic Mostowski Hierarchy and Distance-Parity Automata
Résumé
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 distance-parity 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 $\omega+1$. The uniform universality problem consists in determining if this cost function is bounded by a finite value.