Unambiguous Buchi is weak

TL;DR

This paper proves a complexity collapse for unambiguous nondeterministic parity tree automata, showing their recognized languages are simpler than previously thought, especially for unambiguous Büchi automata which recognize Borel languages.

Contribution

It establishes a new complexity collapse result for unambiguous automata, linking unambiguity with lower descriptive complexity in tree automata.

Findings

Unambiguous automata with certain parity conditions recognize languages in lower complexity classes.

Unambiguous Büchi automata recognize Borel languages.

First known result linking unambiguity to parity index collapse.

Abstract

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 non-deterministic 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 Buchi 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 Buchi 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.