State Complexity of Basic Operations on Finite Languages
The state complexity of basic operations on regular languages has been studied in ,,. Here we focus on finite languages. We show that the catenation of two finite languages accepted by an mstate and an n-state DFA, respectively, with m > n is accepted by a DFA of (m − n + 3)2n−2 − 1 states in the two-letter alphabet case, and this bound is shown to be reachable. We also show that the tight upperbounds for the number of states of a DFA that accepts the star of an n-state finite language is 2n−3 + 2n−4 in the two-letter alphabet case. The same bound for reversal is 3 · 2p−1 − 1 when n is even and 2p − 1 when n is odd. Results for alphabets of an arbitrary size are also obtained. These upper-bounds for finite languages are strictly lower than the corresponding ones for general regular languages.
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