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 $[n{]}^{[n]\times [k]}\times 2^{[n]}$ 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 ${\alpha }_k$.