On the Number of Accepting States of Finite Automata

Abstract

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 $\nasc(R)\leq 2$ for any regular language $R$. Moreover, for a $k$-ary regularity preserving operation $\circ$ on languages, we define $g_{\circ}^{\asc}(n_1,n_2,\dots ,n_k)$ as the set of all integers~$r$ such that there are $k$ regular languages $L_i$, $1\leq i\leq k$, such that $\asc(L_i)=n_i$ for $1\leq i\leq k$ and $\asc(\circ(L_1,L_2,\dots ,L_k))=r$. We determine this set for the operations complement, union, product, Kleene closure, and set difference.