Nondeterministic Finite Automata

A nondeterministic finite accepter or nfa $M$ is defined as quintuple

$$M = (Q, \Sigma, \delta, q_0, F),$$

where

• $Q$ is a finite set of internal states.
• $\Sigma$ is a finite set of symbols, called input alphabet.
• $\delta: Q \times (\Sigma \cup \{ \lambda \}) \rightarrow 2^Q$ is a total function called the transition function.
• $q_0$ is the initial state.
• $F \subseteq Q$ is a set of final states.

Notice that the definition of $\delta^*$ is not same as dfa. $\delta^*(q_i, w)$ contains state $q_j$ if and only if there is a walk in the transition graph from $q_i$ to $q_j$ labelled by $w$. The language accepted by a nfa $M= (Q, \Sigma, \delta, q_0, F)$ is the set of all strings on $\Sigma$ accepted by $M$. Formally,

$$L(M) = \{ w \in \Sigma^*: \delta^*(q_0, w) \cap F \not = \Phi \}.$$