Unambiguous languages exhaust the index hierarchy

This work is a study of the expressive power of unambiguity in the case of infinite trees. An automaton is called unambiguous if it has at most one accepting run on every input, the language of such an automaton is called an unambiguous language. It is known that not every regular language of infinite trees is unambiguous. Except that, very little is known about which regular tree languages are unambiguous. This paper answers the question whether unambiguous languages are of bounded complexity among all regular tree languages. The notion of complexity is the canonical one, called the (parity or Rabin-Mostowski) index hierarchy. The answer is negative, as exhibited by a family of examples of unambiguous languages the cannot be recognised by any alternating parity tree automata of bounded range of priorities. Hardness of the given examples is based on the theory of signatures in parity games, previously studied by Walukiewicz. This theory is further developed here to construct canonical signatures. The technical core of the article is a parity game that compares signatures of a given pair of parity games (without an increase in the index).