Enumeration and random generation of accessible automata

Frédérique Bassino (1) , Cyril Nicaud (2)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

We present a bijection between the set A_n of deterministic and accessible automata with n states on a k-letters alphabet and some diagrams, which can themselves be represented as partitions of a set of kn+1 elements into n non-empty subsets. This combinatorial construction shows that the asymptotic order of the cardinality of A_n is related to Stirling numbers of the second kind . Our bijective approach also yields an efficient random sampler of automata with n states for the uniform distribution: its complexity is O(n^{3/2}), using the framework of Boltzmann samplers.

Keywords: finite automata; bijections \sep asymptotic enumeration; random generation; Boltzmann samplers

Domains: Data Structures and Algorithms [cs.DS]; Combinatorics [math.CO]

Frédérique Bassino, Cyril Nicaud. Enumeration and random generation of accessible automata. Theoretical Computer Science, 2007, 381, pp.86-104. ⟨hal-00459712⟩