Asymptotic enumeration of Minimal Automata

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.

💡 Research Summary

The paper investigates the asymptotic proportion of minimal automata among all accessible deterministic complete automata with n states over a k‑letter alphabet. The authors consider a uniform distribution over transition structures (i.e., each possible k‑map is equally likely) and a Bernoulli distribution with parameter b (0 < b < 1) over the set of final states. The main result is that, for large n, the fraction of minimal automata is

  1 − C(k,b)·n^{−k+2} + o(n^{−k+2}),

where C(k,b) = (1 − 2b(1 − b))·c_k, and c_k = ½·ω_k·k. The constant ω_k is defined implicitly by the transcendental equation

  ω_k = −ln(1 − ω_k)/k,

or equivalently ω_k = 1 + (1/k)·W(−k e^{−k}) using the Lambert‑W function. For the uniform choice of final states (b = ½) the factor simplifies to C(k,½) = c_k.

The proof proceeds by introducing two small subgraph patterns in the transition digraph: an “M‑motif” (two distinct states i and j that share the same outgoing target for every alphabet letter) and a “three‑state M‑motif” (three states sharing all outgoing targets). If an automaton contains an M‑motif and the two states have the same final‑state status, then the two states are Myhill‑Nerode equivalent, making the automaton non‑minimal. The probability that the two states have the same final status under the Bernoulli model is 2b(1 − b).

Random transition structures behave locally like independent k‑maps despite the global accessibility constraint. In such a random k‑map, the indegree distribution of a state is zero with probability zero, and otherwise follows a Poisson distribution with mean k·ω_k. This leads to an explicit expression for the expected number of M‑motifs:

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