Unambiguous Büchi Is Weak

A non-deterministic automaton running on infinite trees is unambiguous if it has at most one accepting run on every tree. The class of languages recognisable by unambiguous tree automata is still not well-understood. In particular, decidability of the problem whether a given language is recognisable by some unambiguous automaton is open. Moreover, there are no known upper bounds on the descriptive complexity of unambiguous languages among all regular tree languages. In this paper we show the following complexity collapse: if a nondeterministic parity tree automaton A is unambiguous and its priorities are between i and 2n then the language recognised by A is in the class Comp(i + 1, 2n). A particular case of this theorem is for i = n = 1: if A is an unambiguous Büchi tree automaton then L(A) is recognisable by a weak alternating automaton (or equivalently definable in weak MSO). The main motivation for this result is a theorem by Finkel and Simonnet stating that every unambiguous Büchi automaton recognises a Borel language. The assumptions of the presented theorem are syntactic (we require one automaton to be both unambiguous and of particular parity index). However, to the authors’ best knowledge this is the first theorem showing a collapse of the parity index that exploits the fact that a given automaton is unambiguous. ∗This work has been supported by National Science Centre grant no. DEC2012/05/N/ST6/03254.