Enumeration and random generation of accessible automata

Abstract

We present a bijection between the set An 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 k n + 1 elements into n non-empty subsets. This combinatorial construction shows that the asymptotic order of the cardinality of An is related to the Stirling number {(k n; n)}. Our bijective approach also yields an efficient random sampler, for the uniform distribution, of automata with n states, its complexity is O (n3 / 2), using the framework of Boltzmann samplers. © 2007 Elsevier Ltd. All rights reserved.