Formal Definition of Non-Deterministic Finite State Machine (NFA)

A nondeterministic finite state automaton (NFA) is a 5-tuple, (S, Σ, T, s0, A), consisting of
 a finite set of states (S)
 a finite set of input symbols (Σ)
 a transition function (T : S × (Σ ∪{ε}) → P(S)).
 an initial (or start) state s0 such that s0 ∈ S
 a set of states A distinguished as accepting (or final) states (A ⊆ S)
where P(S) is the power set of S, ε is the empty string, and Σ is the input symbol alphabet.

Given an NFA M.
Given a string w.
There is any number of computation paths of M with input w.
M accepts w if some computation path ends in a final state.

Formal Definition of Deterministic Finite State Machine (DFA):

Formal Definition of Deterministic Finite State Machine (DFA):

A DFA is a 5-tuple, (S, Σ, T, s, A), consisting of

 a finite set of states (S)
 a finite set called the alphabet (Σ)
 a transition function (T : S × Σ → S)
 a start state (s ∈ S)
 a set of accept states (A ⊆ S)

Given an DFA M.
Given a string w.
There is at most one computation path of M with input w.
M accepts w if that computation path ends in a final state.