The state complexity of random DFAs

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 $[n]<sup>{[n]\times[k]}\times2<sup>{[n]}$ 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 $\alpha_k$.