Asymptotics of Minimal Deterministic Finite Automata Recognizing a Finite Binary Language

We show that the number of minimal deterministic finite automata with n+1 states recognizing a finite binary language grows asymptotically for n ? ? like ?(n! 8? e^{3 a? n^{1/3}} n^{7/8}), where a? ? -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

Succinct representation for (non)deterministic finite automata

International audienceNon)-Deterministic finite automata are one of the simplest models of computation studied in automata theory. Here we study them through the lens of succinct data structures. Towards this goal, we design a data structure for any deterministic automaton D having n states over a σ-letter alphabet using (σ − 1)n log n(1 + o(1)) bits, that determines, given a string x, whether D accepts x in optimal O (|x|) time. We also consider the case when there are N < σ n non-failure transitions, and obtain various time-space trade-offs. Here some of our results are better than the recent work of Cotumaccio and Prezza (SODA 2021). We also exhibit a data structure for non-deterministic automaton N using σ n 2 + n bits that takes O (n 2 |x|) time for string membership checking. Finally, we also provide time and space efficient algorithms for performing several standard operations on the languages accepted by finite automata

Enumeration of Formal Languages

We survey recent results on the enumeration of formal languages. In particular, we consider enumeration of regular languages accepted by deterministic and nondeterministic finite automata with n states, regular languages generated by regular expressions of a fixed length, and ω-regular languages accepted by Müller automata. We also survey the uncomputability of enumeration of context-free languages and more general structures.