Ambiguity Hierarchy of Regular Infinite Tree Languages

Authors Alexander Rabinovich , Doron Tiferet

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Author Details

Alexander Rabinovich
  • Tel Aviv University, Israel
Doron Tiferet
  • Tel Aviv University, Israel


We would like to thank an anonymous referee for pointing out to [Marcin Bilkowski and Michal Skrzypczak, 2013].

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Alexander Rabinovich and Doron Tiferet. Ambiguity Hierarchy of Regular Infinite Tree Languages. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 there is k ∈ ℕ, such that for every input it has at most k accepting computations. 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 ω-words every regular language is accepted by an unambiguous Büchi automaton [Arnold, 1983] and by a deterministic parity automaton. Over infinite trees there are ambiguous languages [Carayol et al., 2010]. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages which are not k-1 ambiguous; there are finitely (respectively countably, uncountably) ambiguous languages which are not boundedly (respectively finitely, countably) ambiguous.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • automata on infinite trees
  • ambiguous automata
  • monadic second-order logic


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