On the Number of Accepting States of Finite Automata

In this paper, we start a systematic study of the number of accepting states. For a regular language L, we define the complexity asc(L) as the minimal number of accepting states necessary to accept L by deterministic finite automata. With respect to nondeterministic automata, the corresponding measure is nasc(L). We prove that, for any non-negative integer n, there is a regular language L such that asc(L) = n, whereas we have nasc(R) ≤ 2 for any regular language R. Moreover, for a k-ary regularity preserving operation ◦ on languages, we define gasc ◦ (n1, n2, . . . , nk) as the set of all integers r such that there are k regular languages Li, 1 ≤ i ≤ k, such that asc(Li) = ni for 1 ≤ i ≤ k and asc(◦(L1, L2, . . . , Lk)) = r. We determine this set for the operations complement, union, product, Kleene closure, and set difference.