Date: 16 Oct 2001

State Complexity of Basic Operations on Finite Languages

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Abstract

The state complexity of basic operations on regular languages has been studied in [9],[10],[11]. 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 (mn + 3)2 n−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 2 n−3 + 2 n−4 in the two-letter alphabet case. The same bound for reversal is 3 · 2 p−1 − 1 when n is even and 2 p − 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.

The work reported here has been supported by the Natural Sciences and Engineering Research Council of Canada Grants OGP0041630 and OGP0147224.