Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers

Frédérique Bassino (1) , Cyril Nicaud (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

We present a bijection between the set $\mathcal{A}_n$ of deterministic and accessible automata with $n$ states on a $k$-letters alphabet and some diagrams, which can themselves be represented as partitions of the set $[\![ 1..(kn+1) ]\!]$ into $n$ non-empty parts. This combinatorial construction shows that the asymptotic order of the cardinality of $\mathcal{A}_n$ is related to the Stirling number $\{^{kn}_n\}$. Our bijective approach also yields an efficient random sampler of automata with $n$ states, of complexity $O(n^{3/2})$, using the framework of Boltzmann samplers.

Mots clés: finite automata; bijection; asymptotic enumeration; random generation; Boltzmann samplers

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Frédérique Bassino, Cyril Nicaud. Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. pp.151-160, ⟨10.46298/dmtcs.3499⟩. ⟨hal-00619870v2⟩