Let $\cal M$ be a finite automaton with transition function $\delta$,
initial state $q_1$, and accepting states $F$.  If $q_i$ is any state of
$\cal M$ and $u \in A^*$, where $A$ is the alphabet of $\cal M$, we shall
write $\delta^*(q_i, u)$ for the state which $\cal M$ will enter if it
begins in state $q_i$ at the left end of the string $u$ and moves across
$u$ until the entire string has been processed.  A formal definition by
recursion is
$$\eqalign{
\delta^*(q_i, 0) &= q_i, \cr
\delta^*(a_i, us_j) &= \delta(\delta^*(q_i, u), s_j).
}$$
Obviously, $\delta^*(q_i, s_j) = \delta(q_i, s_j)$.  Then we say that $\cal
M$ {\it accepts\/} a word $u$ provided that $\delta^*(q_1, u) \in F$.
$\cal M$ {\it rejects\/} $u$ means that $\delta^*(q_1, u) \in Q - F$.
Finally, the {\it language accepted by\/} $\cal M$, written $L({\cal M})$,
is the set of all $u \in A^*$ accepted by $\cal M$:
$$
L({\cal M}) = \{ u \in A^* \mid \delta^*(q_1, u) \in F\}.
$$
\bye