Ambiguity Hierarchy of Regular Infinite Tree Languages

TL;DR

This paper explores the hierarchy of ambiguity levels in regular infinite tree languages, demonstrating that for each ambiguity degree, there are languages that cannot be classified under lower ambiguity levels, revealing a complex structure.

Contribution

It establishes a hierarchy of ambiguity degrees for regular infinite tree languages, showing the existence of languages with strictly increasing ambiguity levels.

Findings

For every k > 1, there are k-ambiguous languages not k-1 ambiguous.

There are languages with various ambiguity degrees that are not boundedly or finitely ambiguous.

The hierarchy of ambiguity levels is strict and infinite for regular infinite tree languages.

Abstract

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k > 0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if it is k-ambiguous for some $k \in \mathbb{N}$. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words every regular language is accepted by a deterministic automaton. Over finite trees every regular language is accepted by an unambiguous automaton. Over $\omega$-words every regular language is accepted by an unambiguous B\"uchi automaton and by a deterministic parity automaton. Over infinite trees Carayol et al. showed that there are ambiguous languages. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages that are not k - 1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages that are not boundedly (respectively finitely, countably) ambiguous.