Kleene closure and state complexity

Palmovský, Matúš

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 50 (2016) no. 3, pp. 251-261.

We prove that the automaton presented by Maslov [Soviet Math. Doklady 11 (1970) 1373–1375] meets the upper bound $3/4·2^n$ on the state complexity of Kleene closure. Our main result shows that the upper bounds $2^{n-1}+2^{n-1-k}$ 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 $1\le k\le n$ 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.

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