Higher-order parity automata

We introduce a notion of higher-order parity automaton which extends to infinitary simply-typed λ-terms the traditional notion of parity tree automaton on infinitary ranked trees. Our main result is that the acceptance of an infinitary λ-term by a higher-order parity automaton A is decidable, whenever the infinitary λ-term is generated by a finite and simply-typed λY-term. The decidability theorem is established by combining ideas coming from linear logic, from denotational semantics and from infinitary rewriting theory.