Weighted Automata and Expressions over Pre-Rational Monoids

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

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Author Details

Nicolas Baudru
  • Aix Marseille Univ, CNRS, LIS, Marseille, France
Louis-Marie Dando
  • Aix Marseille Univ, CNRS, LIS, Marseille, France
Nathan Lhote
  • Aix Marseille Univ, CNRS, LIS, Marseille, France
Benjamin Monmege
  • Aix Marseille Univ, CNRS, LIS, Marseille, France
Pierre-Alain Reynier
  • Aix Marseille Univ, CNRS, LIS, Marseille, France
Jean-Marc Talbot
  • Aix Marseille Univ, CNRS, LIS, Marseille, France

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Weighted Automata and Expressions
  • Inverse Monoids
  • Two-Way Automata


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