Editing Nondeterministic finite automaton (section)

===ε-closure of a state or set of states===

For a state <math>q \in Q</math>, let <math>E(q)</math> denote the set of states that are reachable from <math>q</math> by following ε-transitions in the transition function <math>\delta</math>, i.e.,
<math>p \in E(q)</math> if there is a sequence of states <math>q_1,..., q_k</math> such that
* <math>q_1 = q</math>,
* <math>q_{i+1} \in \delta(q_i, \varepsilon)</math> for each <math>1 \le i < k</math>, and
* <math>q_k = p</math>.

<math>E(q)</math> is known as the '''epsilon closure''', (also '''ε-closure''') of <math>q</math>.

The ε-closure of a set <math>P</math> of states of an NFA is defined as the set of states reachable from any state in <math>P</math> following ε-transitions. Formally, for <math>P \subseteq Q</math>, define <math>E(P) = \bigcup\limits_{q\in P} E(q)</math>.