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DOI: 10.4230/LIPIcs.CSL.2022.6

Weighted Automata and Expressions over Pre-Rational Monoids

Authors: Nicolas Baudru, Louis-Marie Dando, Nathan Lhote, Benjamin Monmege, Pierre-Alain Reynier, and Jean-Marc Talbot

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract

The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature; on one hand, lifting this result to a weighted setting has been widely studied. On the other hand, beyond the free monoid, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalising both the free inverse monoid and graded monoids. The presence of idempotent elements in these pre-rational monoids leads in the weighted setting to consider infinite sums. To handle such sums, we will have to restrict ourselves to rationally additive semirings. Our main result is thus a generalisation of the Kleene theorem for pre-rational monoids and rationally additive semirings. As a corollary, we obtain a class of expressions equivalent to weighted two-way automata, as well as one for tree-walking automata.

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Nicolas Baudru, Louis-Marie Dando, Nathan Lhote, Benjamin Monmege, Pierre-Alain Reynier, and Jean-Marc Talbot. Weighted Automata and Expressions over Pre-Rational Monoids. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baudru_et_al:LIPIcs.CSL.2022.6,
  author =	{Baudru, Nicolas and Dando, Louis-Marie and Lhote, Nathan and Monmege, Benjamin and Reynier, Pierre-Alain and Talbot, Jean-Marc},
  title =	{{Weighted Automata and Expressions over Pre-Rational Monoids}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.6},
  URN =		{urn:nbn:de:0030-drops-157266},
  doi =		{10.4230/LIPIcs.CSL.2022.6},
  annote =	{Keywords: Weighted Automata and Expressions, Inverse Monoids, Two-Way Automata}
}

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DOI: 10.4230/LIPIcs.MFCS.2018.6

On Hadamard Series and Rotating Q-Automata

Authors: Louis-Marie Dando and Sylvain Lombardy

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract

In this paper, we study rotating Q-automata, which are (memoryless) automata with weights in Q, that can read the input tape from left to right several times. We show that the series realized by valid rotating Q-automata are Q-Hadamard series (which are the closure of Q-rational series by pointwise inverse), and that every Q-Hadamard series can be realized by such an automaton. We prove that, although validity of rotating Q-automata is undecidable, the equivalence problem is decidable on rotating Q-automata. Finally, we prove that every valid two-way Q-automaton admits an equivalent rotating Q-automaton. The conversion, which is effective, implies the decidability of equivalence of two-way Q-automata.

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Louis-Marie Dando and Sylvain Lombardy. On Hadamard Series and Rotating Q-Automata. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dando_et_al:LIPIcs.MFCS.2018.6,
  author =	{Dando, Louis-Marie and Lombardy, Sylvain},
  title =	{{On Hadamard Series and Rotating Q-Automata}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.6},
  URN =		{urn:nbn:de:0030-drops-95881},
  doi =		{10.4230/LIPIcs.MFCS.2018.6},
  annote =	{Keywords: Rational series, Hadamard operations, Rotating automata, Two-way automata}
}

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Weighted Automata and Expressions over Pre-Rational Monoids

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On Hadamard Series and Rotating Q-Automata

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