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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

This paper investigates canonical transducers for rational functions over infinite words, i.e., functions of infinite words defined by finite transducers. We first consider sequential functions, defined by finite transducers with a deterministic underlying automaton. We provide a Myhill-Nerode-like characterization, in the vein of Choffrut's result over finite words, from which we derive an algorithm that computes a transducer realizing the function which is minimal and unique (up to the automaton for the domain).
The main contribution of the paper is the notion of a canonical transducer for rational functions over infinite words, extending the notion of canonical bimachine due to Reutenauer and Schützenberger from finite to infinite words. As an application, we show that the canonical transducer is aperiodic whenever the function is definable by some aperiodic transducer, or equivalently, by a first-order transduction. This allows to decide whether a rational function of infinite words is first-order definable.

Emmanuel Filiot, Olivier Gauwin, Nathan Lhote, and Anca Muscholl. On Canonical Models for Rational Functions over Infinite Words. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{filiot_et_al:LIPIcs.FSTTCS.2018.30, author = {Filiot, Emmanuel and Gauwin, Olivier and Lhote, Nathan and Muscholl, Anca}, title = {{On Canonical Models for Rational Functions over Infinite Words}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {30:1--30:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.30}, URN = {urn:nbn:de:0030-drops-99295}, doi = {10.4230/LIPIcs.FSTTCS.2018.30}, annote = {Keywords: transducers, infinite words, minimization, aperiodicty, first-order logic} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

It is known that a language of finite words is definable in monadic second-order logic - MSO - (resp. first-order logic - FO -) iff it is recognized by some finite automaton (resp. some aperiodic finite automaton). Deciding whether an automaton A is equivalent to an aperiodic one is known to be PSPACE-complete. This problem has an important application in logic: it allows one to decide whether a given MSO formula is equivalent to some FO formula. In this paper, we address the aperiodicity problem for functions from finite words to finite words (transductions), defined by finite transducers, or equivalently by bimachines, a transducer model studied by Schützenberger and Reutenauer. Precisely, we show that the problem of deciding whether a given bimachine is equivalent to some aperiodic one is PSPACE-complete.

Emmanuel Filiot, Olivier Gauwin, and Nathan Lhote. Aperiodicity of Rational Functions Is PSPACE-Complete. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{filiot_et_al:LIPIcs.FSTTCS.2016.13, author = {Filiot, Emmanuel and Gauwin, Olivier and Lhote, Nathan}, title = {{Aperiodicity of Rational Functions Is PSPACE-Complete}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.13}, URN = {urn:nbn:de:0030-drops-68482}, doi = {10.4230/LIPIcs.FSTTCS.2016.13}, annote = {Keywords: Rational word transductions, decision problem, aperiodicity, bimachines} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We consider minimization problems for natural parameters of word transducers: the number of passes performed by two-way transducers and the number of registers used by streaming transducers. We show how to compute in ExpSpace the minimum number of passes needed to implement a transduction given as sweeping transducer, and we provide effective constructions of transducers of (worst-case optimal) doubly exponential size. We then consider streaming transducers where concatenations of registers are forbidden in the register updates. Based on a correspondence between the number of passes of sweeping transducers and the number of registers of equivalent concatenation-free streaming transducers, we derive a minimization procedure for the number of registers of concatenation-free streaming transducers.

Félix Baschenis, Olivier Gauwin, Anca Muscholl, and Gabriele Puppis. Minimizing Resources of Sweeping and Streaming String Transducers. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 114:1-114:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{baschenis_et_al:LIPIcs.ICALP.2016.114, author = {Baschenis, F\'{e}lix and Gauwin, Olivier and Muscholl, Anca and Puppis, Gabriele}, title = {{Minimizing Resources of Sweeping and Streaming String Transducers}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {114:1--114:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.114}, URN = {urn:nbn:de:0030-drops-62496}, doi = {10.4230/LIPIcs.ICALP.2016.114}, annote = {Keywords: word transducers, streaming, 2-way, sweeping transducers, minimization} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Two-way finite-state transducers on words are strictly more expressive than one-way transducers. It has been shown recently how to decide if a two-way functional transducer has an equivalent one-way transducer, and the complexity of the algorithm is non-elementary. We propose an alternative and simpler characterization for sweeping functional transducers, namely, for transducers that can only reverse their head direction at the extremities of the input. Our algorithm works in 2EXPSPACE and, in the positive case, produces an equivalent one-way transducer of doubly exponential size. We also show that the bound on the size of the transducer is tight, and that the one-way definability problem is undecidable for (sweeping) non-functional transducers.

Félix Baschenis, Olivier Gauwin, Anca Muscholl, and Gabriele Puppis. One-way Definability of Sweeping Transducer. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 178-191, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{baschenis_et_al:LIPIcs.FSTTCS.2015.178, author = {Baschenis, F\'{e}lix and Gauwin, Olivier and Muscholl, Anca and Puppis, Gabriele}, title = {{One-way Definability of Sweeping Transducer}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {178--191}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.178}, URN = {urn:nbn:de:0030-drops-56297}, doi = {10.4230/LIPIcs.FSTTCS.2015.178}, annote = {Keywords: Regular word transductions, sweeping transducers, one-way definability} }

Document

**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We consider the problem of evaluating in streaming (i.e. in a single left-to-right pass) a nested word transduction with a limited amount of memory. A transduction T is said to be height bounded memory (HBM) if it can be evaluated with a memory that depends only on the size of T and on the height of the input word. We show that it is decidable in coNPTime for a nested word transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In this case, the required amount of memory may depend exponentially on the height of the word. We exhibit a sufficient, decidable condition for a VPT to be evaluated with a memory that depends quadratically on the height of the word. This condition defines a class of transductions that strictly contains all determinizable VPTs.

Emmanuel Filiot, Olivier Gauwin, Pierre-Alain Reynier, and Frédéric Servais. Streamability of Nested Word Transductions. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 312-324, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{filiot_et_al:LIPIcs.FSTTCS.2011.312, author = {Filiot, Emmanuel and Gauwin, Olivier and Reynier, Pierre-Alain and Servais, Fr\'{e}d\'{e}ric}, title = {{Streamability of Nested Word Transductions}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {312--324}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.312}, URN = {urn:nbn:de:0030-drops-33523}, doi = {10.4230/LIPIcs.FSTTCS.2011.312}, annote = {Keywords: nested word, visibly pushdown transducer, streaming, XML} }

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