The state complexity of random DFAs

TL;DR

This paper investigates the typical number of states in the minimal automaton for random DFAs, revealing that it concentrates around a linear function of the number of states with high probability.

Contribution

It provides a probabilistic analysis of the state complexity of random DFAs, establishing an explicit asymptotic formula for the minimal automaton size.

Findings

The minimal DFA size is approximately _k n with high probability.

The size concentrates around the linear term with a standard deviation of order ( ext{n}\log n).

The result applies to uniform random DFAs over a fixed alphabet.

Abstract

The state complexity of a Deterministic Finite-state automaton (DFA) is the number of states in its minimal equivalent DFA. We study the state complexity of random $n$-state DFAs over a $k$-symbol alphabet, drawn uniformly from the set $[n]^{[n]\times[k]}\times2^{[n]}$ of all such automata. We show that, with high probability, the latter is $\alpha_k n + O(\sqrt n\log n)$ for a certain explicit constant $\alpha_k$.