Ambiguity Hierarchy of Regular Infinite Tree Languages
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.
automata on infinite trees
ambiguous automata
monadic second-order logic
Theory of computation~Automata over infinite objects
80:1-80:14
Regular Paper
Supported in part by Len Blavatnik and the Blavatnik Family foundation.
We would like to thank an anonymous referee for pointing out to [Marcin Bilkowski and Michal Skrzypczak, 2013].
Alexander
Rabinovich
Alexander Rabinovich
Tel Aviv University, Israel
https://www.cs.tau.ac.il/~rabinoa/
https://orcid.org/0000-0002-1460-2358
Doron
Tiferet
Doron Tiferet
Tel Aviv University, Israel
10.4230/LIPIcs.MFCS.2020.80
André Arnold. Rational omega-languages are non-ambiguous. Theor. Comput. Sci., 26:221-223, September 1983. URL: https://doi.org/10.1016/0304-3975(83)90086-5.
https://doi.org/10.1016/0304-3975(83)90086-5
Vince Bárány, Łukasz Kaiser, and Alex Rabinovich. Expressing cardinality quantifiers in monadic second-order logic over trees. Fundamenta Informaticae, 100(1-4):1-17, 2010.
Marcin Bilkowski and Michal Skrzypczak. Unambiguity and uniformization problems on infinite trees. In Simona Ronchi Della Rocca, editor, Computer Science Logic 2013 (CSL 2013), CSL 2013, September 2-5, 2013, Torino, Italy, volume 23 of LIPIcs, pages 81-100. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2013. URL: https://doi.org/10.4230/LIPIcs.CSL.2013.81.
https://doi.org/10.4230/LIPIcs.CSL.2013.81
Arnaud Carayol and Christof Löding. MSO on the infinite binary tree: Choice and order. In International Workshop on Computer Science Logic, pages 161-176. Springer, 2007.
Arnaud Carayol, Christof Löding, Damian Niwinski, and Igor Walukiewicz. Choice functions and well-orderings over the infinite binary tree. Open Mathematics, 8(4):662-682, 2010.
E Allen Emerson and Charanjit S Jutla. Tree automata, mu-calculus and determinacy. In FoCS, volume 91, pages 368-377. Citeseer, 1991.
Yuri Gurevich and Leo Harrington. Trees, automata, and games. In Proceedings of the fourteenth annual ACM symposium on Theory of computing, pages 60-65, 1982.
Yuri Gurevich and Saharon Shelah. Rabin’s uniformization problem 1. The Journal of Symbolic Logic, 48(4):1105-1119, 1983.
Damian Niwiński. On the cardinality of sets of infinite trees recognizable by finite automata. In Andrzej Tarlecki, editor, Mathematical Foundations of Computer Science 1991, pages 367-376, Berlin, Heidelberg, 1991. Springer Berlin Heidelberg.
D. Perrin and J.É. Pin. Infinite Words: Automata, Semigroups, Logic and Games. ISSN. Elsevier Science, 2004. URL: https://books.google.fr/books?id=S7hHhJc4iNgC.
https://books.google.fr/books?id=S7hHhJc4iNgC
Michael O Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of the american Mathematical Society, 141:1-35, 1969.
Wolfgang Thomas. Automata on infinite objects. In Formal Models and Semantics, pages 133-191. Elsevier, 1990.
Boris A. Trakhtenbrot and Ya. M. Barzdin. Finite automata, behavior and synthesis. 1973.
Alexander Rabinovich and Doron Tiferet
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode