Deterministic Finite Automaton

A deterministic finite automaton M is a 5-tuple, (Q, Σ, δ, q₀, F), consisting of

• a finite set of states (Q)
• a finite set of input symbols called the alphabet (Σ)
• a transition function (δ : Q × Σ → Q)
• a start state (q₀ ∈ Q)
• a set of accept states (F ⊆ Q)

Let w = a₁a₂ ... a_n be a string over the alphabet Σ. The automaton M accepts the string w if a sequence of states, r₀,r₁, ..., r_n, exists in Q with the following conditions:

1. r₀ = q₀
2. r_i+1 = δ(r_i, a_i+1), for i = 0, ..., n−1
3. r_n ∈ F.

In words, the first condition says that the machine starts in the start state q₀. The second condition says that given each character of string w, the machine will transition from state to state according to the transition function δ. The last condition says that the machine accepts w if the last input of w causes the machine to halt in one of the accepting states. Otherwise, it is said that the automaton rejects the string. The set of strings M accepts is the language recognized by M and this language is denoted by L(M).

A deterministic finite automaton without accept states and without a starting state is known as a transition system or semiautomaton.

For more comprehensive introduction of the formal definition see automata theory.