Deterministic Finite Automata

A deterministic finite accepter or dfa $M$ 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.
• $\delta: Q \times \Sigma \rightarrow Q$ is a total function called the transition function.
• $q_0$ is the initial state.
• $F \subseteq Q$ is a set of final states.

It is convenient to introduce the extended transition function

$$\delta^*: Q \times \Sigma^* \rightarrow Q.$$

The second argument of $\delta^*$ is a string rather than a single symbol, and its value gives the state the automaton will be in after reading that string. The language accepted by a dfa $M= (Q, \Sigma, \delta, q_0, F)$ is the set of all strings on $\Sigma$ accepted by $M$. Formally,

$$L(M) = \{ w \in \Sigma^*: \delta^*(q_0, w) \in F \}.$$