Automata Implementation

Volume 2214 of the series Lecture Notes in Computer Science pp 60-70


State Complexity of Basic Operations on Finite Languages

  • C. CâmpeanuAffiliated withFundamentals of Computer Science Department, Faculty of Mathematics University of Bucharest
  • , K. CulikIIAffiliated withDepartment of Computer Science, University of South Carolina
  • , Kai SalomaaAffiliated withDepartment of Computer Science, The University of Western Ontario
  • , Sheng YuAffiliated withDepartment of Computer Science, The University of Western Ontario

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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)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.