Kleene Closure and State Complexity

We prove that the automaton presented by Maslov [Soviet Math. Doklady 11, 1373–1375 (1970)] meets the upper bound 3/4 · 2n on the state complexity of Kleene closure. This fixes a small error in this paper that claimed the upper bound 3/4 · 2n − 1. 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 in the binary case. We also present some results of our calculations. We consider not only the worst case, but we study all possible values that can be obtained as the state complexity of Kleene closure of a regular language accepted by a minimal n-state DFA. Using the lists of pairwise non-isomorphic binary automata of 2,3,4, and 5 states, we compute the frequencies of the resulting complexities for Kleene closure, and show that every value in the range from 1 to 3/4 ·2n occurs at least ones. In the case of n = 6,7,8, we change the strategy, and consider binary automata, in which the first symbol is a circular shift of the states, and the second symbol is generated randomly. We show that all values from 1 to 3/4 · 2n are attainable, that is, for every m with 1≤m≤ 3/4 ·2n, there exists an nstate binary DFA A such that the state complexity of L(A)∗ is exactly m.