Square, Power, Positive Closure, and Complementation on Star-Free Languages

Abstract

We examine the deterministic and nondeterministic state complexity of square, power, positive closure, and complementation on star-free languages. For the state complexity of square, we get a non-trivial upper bound (n−1)2n−2(n−2) and a lower bound of order Θ(2n). For the state complexity of the k-th power in the unary case, we get the tight upper bound k(n − 1) + 1. Next, we show that the upper bound kn on the nondeterministic state complexity of the k-th power is met by a binary star-free language, while in the unary case, we have a lower bound k(n − 1) + 1. For the positive closure, we show that the deterministic upper bound 2n−1 + 2n−2 − 1, as well as the nondeterministic upper bound n, can be met by star-free languages. We also show that in the unary case, the state complexity of positive closure is n2 − 7n + 13, and the nondeterministic state complexity of complementation is between (n − 1)2 + 1 and n2 − 2.