Kleene closure and state complexity

Abstract

We prove that the automaton presented by Maslov [Soviet Math. Doklady 11 (1970) 1373–1375] meets the upper bound 3/4·2n on the state complexity of Kleene closure. Our main result shows that the upper bounds 2n − 1 + 2n − 1 − k on the state complexity of Kleene closure of a language accepted by an n -state DFA with k final states are tight for every k with 1 ≤ k ≤ n in the binary case. We also study Kleene Closure on prefix-free languages. In the case of prefix-free languages, the Kleene closure may attain just three possible complexities n − 2,n − 1, and n . We give some survey of our computations.