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# Asymptotics of Minimal Deterministic Finite Automata Recognizing a Finite Binary Language

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LIPIcs.AofA.2020.11.pdf
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## 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 As

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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