 Solution

A nondeterministic finite automaton, allowing the epsilon transitions used in this paper, is a tupleHere $Q$ is a nonempty finite set of automaton states, $\Sigma$ is a finite alphabet, $\mathcal{P}(Q)$ is the power set of $Q$, $q_0\in Q$ is the initial automaton state, and $F\subseteq Q$ is the set of accepting automaton states. The symbol $\epsilon \notin \Sigma$ denotes the empty word. Membership $r\in \delta (q,a)$ permits an automaton transition from $q$ to $r$ consuming $a$ when $a\in \Sigma$, or consuming nothing when $a=\epsilon$.

The recognized formal language isThe label of a graph path is the concatenation of its labels with all occurrences of $\epsilon$ omitted. Thus acceptance requires consuming the entire input word; a single successful choice of automaton transitions suffices. A path of length zero is allowed, so the empty word is accepted whenever $q_0\in F$, or an accepting automaton state is reachable by epsilon transitions. If epsilon transitions are excluded from the definition of a nondeterministic finite automaton, the displayed version is called an Epsilon-NFA.