Asymptotics of Minimal Deterministic Finite Automata Recognizing a Finite Binary Language

Authors Andrew Elvey Price , Wenjie Fang , Michael Wallner

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Author Details

Andrew Elvey Price
  • Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, UMR 5800, 351 Cours de la Libération, 33405 Talence Cedex, France
  • Université de Tours, Institut Denis Poisson, UMR 7013, Parc de Grandmont, 37200 Tours, France
Wenjie Fang
  • Laboratoire d'Informatique Gaspard-Monge, UMR 8049, Université Gustave Eiffel, CNRS, ESIEE Paris, 77454 Marne-la-Vallée, France
Michael Wallner
  • Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, UMR 5800, 351 Cours de la Libération, 33405 Talence Cedex, France
  • TU Wien, Institute for Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8 - 10, 1040 Wien, Austria


We would like to thank Cyril Banderier, Tony Guttmann, and Andrea Sportiello for interesting discussions on the presence of a stretched exponential. We also thank our referees for their careful reading and helpful comments.

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Andrew Elvey Price, Wenjie Fang, and Michael Wallner. Asymptotics of Minimal Deterministic Finite Automata Recognizing a Finite Binary Language. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We show that the number of minimal deterministic finite automata with n+1 states recognizing a finite binary language grows asymptotically for n → ∞ like Θ(n! 8ⁿ e^{3 a₁ n^{1/3}} n^{7/8}), where a₁ ≈ -2.338 is the largest root of the Airy function. For this purpose, we use a new asymptotic enumeration method proposed by the same authors in a recent preprint (2019). We first derive a new two-parameter recurrence relation for the number of such automata up to a given size. Using this result, we prove by induction tight bounds that are sufficiently accurate for large n to determine the asymptotic form using adapted Netwon polygons.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
  • Airy function
  • asymptotics
  • directed acyclic graphs
  • Dyck paths
  • bijection
  • stretched exponential
  • compacted trees
  • minimal automata
  • finite languages


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