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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Regular functions of infinite words are (partial) functions realized by deterministic two-way transducers with infinite look-ahead. Equivalently, Alur et. al. have shown that they correspond to functions realized by deterministic Muller streaming string transducers, and to functions defined by MSO-transductions. Regular functions are however not computable in general (for a classical extension of Turing computability to infinite inputs), and we consider in this paper the class of deterministic regular functions of infinite words, realized by deterministic two-way transducers without look-ahead. We prove that it is a well-behaved class of functions: they are computable, closed under composition, characterized by the guarded fragment of MSO-transductions, by deterministic Büchi streaming string transducers, by deterministic two-way transducers with finite look-ahead, and by finite compositions of sequential functions and one fixed basic function called map-copy-reverse.

Olivier Carton, Gaëtan Douéneau-Tabot, Emmanuel Filiot, and Sarah Winter. Deterministic Regular Functions of Infinite Words. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 121:1-121:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{carton_et_al:LIPIcs.ICALP.2023.121, author = {Carton, Olivier and Dou\'{e}neau-Tabot, Ga\"{e}tan and Filiot, Emmanuel and Winter, Sarah}, title = {{Deterministic Regular Functions of Infinite Words}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {121:1--121:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.121}, URN = {urn:nbn:de:0030-drops-181733}, doi = {10.4230/LIPIcs.ICALP.2023.121}, annote = {Keywords: infinite words, streaming string transducers, two-way transducers, monadic second-order logic, look-aheads, factorization forests} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

In this paper, we establish a strong link between the ambiguity for finite words of a Büchi automaton and the ambiguity for infinite words of the same automaton. This link is based on measure theory. More precisely, we show that such an automaton is unambiguous, in the sense that no finite word labels two runs with the same starting state and the same ending state if and only if for each state, the set of infinite sequences labelling two runs starting from that state has measure zero. The measure used to define these negligible sets, that is sets of measure zero, can be any measure computed by a weighted automaton which is compatible with the Büchi automaton. This latter condition is very natural: the measure must only put weight on sets wA^ℕ where w is the label of some run in the Büchi automaton.

Olivier Carton. Ambiguity Through the Lens of Measure Theory. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{carton:LIPIcs.FSTTCS.2022.34, author = {Carton, Olivier}, title = {{Ambiguity Through the Lens of Measure Theory}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.34}, URN = {urn:nbn:de:0030-drops-174269}, doi = {10.4230/LIPIcs.FSTTCS.2022.34}, annote = {Keywords: ambiguity, B\"{u}chi automata, measure theory} }

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

A word-to-word function is rational if it can be realized by a non-deterministic one-way transducer. Over finite words, it is a classical result that any rational function is regular, i.e. it can be computed by a deterministic two-way transducer, or equivalently, by a deterministic streaming string transducer (a one-way automaton which manipulates string registers).
This result no longer holds for infinite words, since a non-deterministic one-way transducer can guess, and check along its run, properties such as infinitely many occurrences of some pattern, which is impossible for a deterministic machine. In this paper, we identify the class of rational functions over infinite words which are also computable by a deterministic two-way transducer. It coincides with the class of rational functions which are continuous, and this property can thus be decided. This solves an open question raised in a previous paper of Dave et al.

Olivier Carton and Gaëtan Douéneau-Tabot. Continuous Rational Functions Are Deterministic Regular. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{carton_et_al:LIPIcs.MFCS.2022.28, author = {Carton, Olivier and Dou\'{e}neau-Tabot, Ga\"{e}tan}, title = {{Continuous Rational Functions Are Deterministic Regular}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {28:1--28:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.28}, URN = {urn:nbn:de:0030-drops-168268}, doi = {10.4230/LIPIcs.MFCS.2022.28}, annote = {Keywords: infinite words, rational functions, determinization, continuity, streaming string transducers, two-way transducers} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

We prove that the Büchi topology, the automatic topology, the alphabetic topology and the strong alphabetic topology are Polish, and provide consequences of this.

Olivier Carton, Olivier Finkel, and Dominique Lecomte. Polishness of Some Topologies Related to Automata. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{carton_et_al:LIPIcs.CSL.2017.22, author = {Carton, Olivier and Finkel, Olivier and Lecomte, Dominique}, title = {{Polishness of Some Topologies Related to Automata}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.22}, URN = {urn:nbn:de:0030-drops-76728}, doi = {10.4230/LIPIcs.CSL.2017.22}, annote = {Keywords: Automata and formal languages; logic in computer science; infinite words; B\"{u}chi automaton; regular omega-language; Cantor space; finer topologies; B\"{u}c} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

A word-to-word function is continuous for a class of languages V if its inverse maps V languages to V. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes.
Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. Previous algebraic studies of transducers have focused on the structure of the underlying input automaton, disregarding the output. We propose a comparison of the two algebraic approaches through two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?

Michaël Cadilhac, Olivier Carton, and Charles Paperman. Continuity and Rational Functions. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 115:1-115:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cadilhac_et_al:LIPIcs.ICALP.2017.115, author = {Cadilhac, Micha\"{e}l and Carton, Olivier and Paperman, Charles}, title = {{Continuity and Rational Functions}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {115:1--115:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.115}, URN = {urn:nbn:de:0030-drops-74583}, doi = {10.4230/LIPIcs.ICALP.2017.115}, annote = {Keywords: Transducers, rational functions, language varieties, continuity} }

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**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Deterministic two-way transducers on finite words have been shown by Engelfriet and Hoogeboom to have the same expressive power as MSO-transductions. We introduce a notion of aperiodicity for these transducers and we show that aperiodic transducers correspond exactly to FO-transductions. This lifts to transducers the classical equivalence for languages between FO-definability, recognition by aperiodic monoids and acceptance by counter-free automata.

Olivier Carton and Luc Dartois. Aperiodic Two-way Transducers and FO-Transductions. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 160-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{carton_et_al:LIPIcs.CSL.2015.160, author = {Carton, Olivier and Dartois, Luc}, title = {{Aperiodic Two-way Transducers and FO-Transductions}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {160--174}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.160}, URN = {urn:nbn:de:0030-drops-54133}, doi = {10.4230/LIPIcs.CSL.2015.160}, annote = {Keywords: Transducer, first-order, two-way, transition monoid, aperiodic} }

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

We prove that for the class of sets of words indexed by countable scattered linear orderings, there is an equivalence between definability in first-order logic, star-free expressions with marked product, and recognizability by finite aperiodic semigroups which satisfy some additional equation.

Alexis Bès and Olivier Carton. Algebraic Characterization of FO for Scattered Linear Orderings. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 67-81, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{bes_et_al:LIPIcs.CSL.2011.67, author = {B\`{e}s, Alexis and Carton, Olivier}, title = {{Algebraic Characterization of FO for Scattered Linear Orderings}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {67--81}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.67}, URN = {urn:nbn:de:0030-drops-32233}, doi = {10.4230/LIPIcs.CSL.2011.67}, annote = {Keywords: linear orderings, first-order logic, semigroups, rational sets} }

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