Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series

Authors Alin Bostan, Arnaud Carayol, Florent Koechlin, Cyril Nicaud

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Alin Bostan
  • INRIA Saclay Île-de-France, Palaiseau, France
Arnaud Carayol
  • LIGM, Univ. Gustave Eiffel, CNRS, Marne-la-Vallée, France
Florent Koechlin
  • LIGM, Univ. Gustave Eiffel, CNRS, Marne-la-Vallée, France
Cyril Nicaud
  • LIGM, Univ. Gustave Eiffel, CNRS, Marne-la-Vallée, France

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Alin Bostan, Arnaud Carayol, Florent Koechlin, and Cyril Nicaud. Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 114:1-114:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and Massazza: weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines, and their multivariate generating series are holonomic. We then show that the converse is not true: we construct a language whose generating series is algebraic (thus holonomic), but which is inherently weakly-ambiguous as a Parikh automata language. Finally, we prove an effective decidability result for the inclusion problem for weakly-unambiguous Parikh automata, and provide an upper-bound on its complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Mathematics of computing → Generating functions
  • generating series
  • holonomicity
  • ambiguity
  • reversal bounded counter machine
  • Parikh automata


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