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Asymptotics of Minimal Deterministic Finite Automata Recognizing a Finite Binary Language

Authors Andrew Elvey Price , Wenjie Fang , Michael Wallner



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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

Acknowledgements

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.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.AofA.2020.11

Abstract

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
Keywords
  • Airy function
  • asymptotics
  • directed acyclic graphs
  • Dyck paths
  • bijection
  • stretched exponential
  • compacted trees
  • minimal automata
  • finite languages

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References

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