Asymptotic enumeration of Minimal Automata

TL;DR

This paper calculates the asymptotic proportion of minimal automata among accessible deterministic automata, revealing that most automata are minimal as the number of states grows, with explicit formulas involving complex equations.

Contribution

It provides an explicit asymptotic enumeration of minimal automata, including a formula for the proportion involving a transcendental equation.

Findings

A fraction ~ 1-C(k,b) n^{-k+2} of automata are minimal

Explicit formula for C(k,b) involving a transcendental equation

Most automata are minimal as the number of states increases

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.