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.
- 1.J.A. Brzozowski, “Canonical regular expressions and minimal state graphs for definite events”, Mathematical Theory of Automata, vol. 12 of MRI Symposia Series, Polytechnic Press, NY, 1962, 529–561.Google Scholar
- 2.C. Câmpeanu, “Regular languages and programming languages”, Revue Roumaine de Linguistique-CLTA, 23 (1986), 7–10.Google Scholar
- 7.B.W. Watson, Taxonomies and Toolkits of Regular Language Algorithms, PhD Dissertation, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1995.Google Scholar
- 8.D. Wood, Theory of Computation, John Wiley and Sons, 1987.Google Scholar
- 11.S. Yu, Regular Languages, in: Handbook of Formal Languages, Vol. I, G. Rozenberg and A. Salomaa eds., Springer Verlag, 1997, pp. 41–110.Google Scholar