  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 $\alpha_k$.
