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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

A function f uniformizes a relation R(X,Y) if R(X,f(X)) holds for every X in the domain of R. The uniformization problem for a logic L asks whether for every L-definable relation there is an L-definable function that uniformizes it. Gurevich and Shelah proved that no Monadic Second-Order (MSO) definable function uniformizes relation "Y is a one element subset of X" in the full binary tree. In other words, there is no MSO definable choice function in the full binary tree.
The cross-section of a relation R(X,Y) at D is the set of all E such that R(D,E) holds. Hence, a function that uniformizes R chooses one element from every non-empty cross-section. The relation "Y is a one element subset of X" has finite and countable cross-sections.
We prove that in the full binary tree the following theorems hold:
▶ Theorem (Finite cross-sections) If every cross-section of an MSO definable relation is finite, then it has an MSO definable uniformizer.
▶ Theorem (Uncountable cross-section) There is an MSO definable relation R such that every MSO definable relation included in R and with the same domain as R has an uncountable cross-section.

Alexander Rabinovich. On Uniformization in the Full Binary Tree. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{rabinovich:LIPIcs.MFCS.2022.77, author = {Rabinovich, Alexander}, title = {{On Uniformization in the Full Binary Tree}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {77:1--77:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.77}, URN = {urn:nbn:de:0030-drops-168757}, doi = {10.4230/LIPIcs.MFCS.2022.77}, annote = {Keywords: Monadic Second-Order Logic, Uniformization} }

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**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. An automaton is boundedly ambiguous if there is k ∈ ℕ, such that for every input it has at most k accepting computations. We consider Parity Tree Automata (PTA) and prove that the problem whether a PTA is not unambiguous (respectively, is not boundedly ambiguous, not finitely ambiguous) is co-NP complete, and the problem whether a PTA is not countably ambiguous is co-NP hard.

Alexander Rabinovich and Doron Tiferet. Degrees of Ambiguity for Parity Tree Automata. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{rabinovich_et_al:LIPIcs.CSL.2021.36, author = {Rabinovich, Alexander and Tiferet, Doron}, title = {{Degrees of Ambiguity for Parity Tree Automata}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {36:1--36:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.36}, URN = {urn:nbn:de:0030-drops-134709}, doi = {10.4230/LIPIcs.CSL.2021.36}, annote = {Keywords: automata on infinite trees, degree of ambiguity, omega word automata, parity automata} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 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.

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)

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@InProceedings{rabinovich_et_al:LIPIcs.MFCS.2020.80, author = {Rabinovich, Alexander and Tiferet, Doron}, title = {{Ambiguity Hierarchy of Regular Infinite Tree Languages}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {80:1--80:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.80}, URN = {urn:nbn:de:0030-drops-127495}, doi = {10.4230/LIPIcs.MFCS.2020.80}, annote = {Keywords: automata on infinite trees, ambiguous automata, monadic second-order logic} }

Document

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. An automaton is boundedly ambiguous if there is k in N, such that for every input it has at most k accepting computations. We consider nondeterministic Büchi automata (NBA) over infinite trees and prove that it is decidable in polynomial time, whether an automaton is unambiguous, boundedly ambiguous, finitely ambiguous, or countably ambiguous.

Alexander Rabinovich and Doron Tiferet. Degrees of Ambiguity of Büchi Tree Automata. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{rabinovich_et_al:LIPIcs.FSTTCS.2019.50, author = {Rabinovich, Alexander and Tiferet, Doron}, title = {{Degrees of Ambiguity of B\"{u}chi Tree Automata}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {50:1--50:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.50}, URN = {urn:nbn:de:0030-drops-116122}, doi = {10.4230/LIPIcs.FSTTCS.2019.50}, annote = {Keywords: automata on infinite trees, ambiguous automata} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

We provide a simple proof of Kamp's theorem.

Alexander Rabinovich. A Proof of Kamp's theorem. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 516-527, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{rabinovich:LIPIcs.CSL.2012.516, author = {Rabinovich, Alexander}, title = {{A Proof of Kamp's theorem}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {516--527}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.516}, URN = {urn:nbn:de:0030-drops-36945}, doi = {10.4230/LIPIcs.CSL.2012.516}, annote = {Keywords: Temporal Logic, Monadic Logic, Expressive Completeness} }

Document

**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Church's Problem asks for the construction of a procedure which, given a logical specification S(I,O) between input strings I and output strings O, determines whether there exists an operator F that implements the specification in the sense that S(I,F(I)) holds for all inputs I. Buechi and Landweber gave a procedure to solve Church's problem for MSO specifications and operators computable by finite-state automata.
We consider extensions of Church's problem in two orthogonal directions: (i) we address the problem in a more general logical setting, where not only the specifications but also the solutions are presented in a logical system; (ii) we consider not only the canonical discrete time domain of the natural numbers, but also the continuous domain of reals.
We show that for every fixed bounded length interval of the reals, Church's problem is decidable when specifications and implementations are described in the monadic second-order logics over the reals with order and the +1 function.

Mark Jenkins, Joël Ouaknine, Alexander Rabinovich, and James Worrell. The Church Synthesis Problem with Metric. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 307-321, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{jenkins_et_al:LIPIcs.CSL.2011.307, author = {Jenkins, Mark and Ouaknine, Jo\"{e}l and Rabinovich, Alexander and Worrell, James}, title = {{The Church Synthesis Problem with Metric}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {307--321}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.307}, URN = {urn:nbn:de:0030-drops-32390}, doi = {10.4230/LIPIcs.CSL.2011.307}, annote = {Keywords: Church's Problem, monadic logic, games, uniformization} }

Document

**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Church's Problem asks for the construction of a procedure which,
given a logical specification $\varphi$ on sequence pairs, realizes
for any input sequence $I$ an output sequence $O$ such that $(I,O)$
satisfies $\varphi$. McNaughton reduced Church's Problem to a problem about two-player$\omega$-games.
B\"uchi and Landweber gave a solution for
Monadic Second-Order Logic of Order ($\MLO$) specifications in terms of finite-state strategies.
We consider two natural generalizations of the Church problem to
countable ordinals: the first deals with finite-state strategies;
the second deals with $\MLO$-definable strategies. We investigate
games of arbitrary countable length and prove the computability of
these generalizations of Church's problem.

Alexander Rabinovich. Synthesis of Finite-state and Definable Winning Strategies. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 359-370, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{rabinovich:LIPIcs.FSTTCS.2009.2332, author = {Rabinovich, Alexander}, title = {{Synthesis of Finite-state and Definable Winning Strategies}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {359--370}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2332}, URN = {urn:nbn:de:0030-drops-23320}, doi = {10.4230/LIPIcs.FSTTCS.2009.2332}, annote = {Keywords: Games of ordinal length, Church Synthesis Problem, Monadic Logic, Composition Method} }

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