TY - GEN

T1 - Ambiguity hierarchy of regular infinite tree languages

AU - Rabinovich, Alexander

AU - Tiferet, Doron

N1 - Publisher Copyright:
© Nathalie Bertrand; licensed under Creative Commons License CC-BY 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020).

PY - 2020/8/1

Y1 - 2020/8/1

N2 - 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.

AB - 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.

KW - Ambiguous automata

KW - Automata on infinite trees

KW - Monadic second-order logic

UR - http://www.scopus.com/inward/record.url?scp=85090507225&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2020.80

DO - 10.4230/LIPIcs.MFCS.2020.80

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AN - SCOPUS:85090507225

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020

A2 - Esparza, Javier

A2 - Kral�, Daniel

A2 - Kral�, Daniel

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020

Y2 - 25 August 2020 through 26 August 2020

ER -