Asymptotic enumeration of Minimal Automata

Abstract

We determine the asymptotic proportion of minimal automata, within n-state accessible deterministic complete automata over a k-letter alphabet, with the uniform distribution over the possible transition structures, and a binomial distribution over terminal states, with arbitrary parameter b. It turns out that a fraction ~ 1-C(k,b) n^{-k+2} of automata is minimal, with C(k,b) a function, explicitly determined, involving the solution of a transcendental equation.