A gap property of deterministic tree languages

We show that a tree language recognized by a deterministic parity automaton is either hard for the co–Büchi level and therefore cannot be recognized by a weak alternating automaton, or is on a very low level in the hierarchy of weak alternating automata. A topological counterpart of this property is that a deterministic tree language is either Π 1 complete (and hence non Borel), or it is on the level Π 3 of the Borel hierarchy. We also give a new simple proof of the strictness of the hierarchy of weak alternating automata. For Anatol Slissenko at his 60th birthday.