Ambiguity hierarchy of regular infinite tree languages

Alexander Rabinovich, Doron Tiferet

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 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 kambiguous (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 [1] and by a deterministic parity automaton. Over infinite trees there are ambiguous languages [5]. 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.

Original languageEnglish
Title of host publication45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
EditorsJavier Esparza, Daniel Kral�, Daniel Kral�
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771597
DOIs
StatePublished - 1 Aug 2020
Event45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020 - Prague, Czech Republic
Duration: 25 Aug 202026 Aug 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume170
ISSN (Print)1868-8969

Conference

Conference45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
Country/TerritoryCzech Republic
CityPrague
Period25/08/2026/08/20

Keywords

  • Ambiguous automata
  • Automata on infinite trees
  • Monadic second-order logic

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