2.1: Deterministic Finite Automata

Configuration Function

A function ⌒ : K × ∑⁺ → K × ∑*. If (q,w) ∈ K × ∑⁺ and (q',w') ∈ K × ∑*, then (q,w) ⌒ (q',w') if and only if w = aw' for some a ∈ ∑, and δ(q,w) = q'. Let ⌒* denote the reflexive, transitive closure of ⌒.

Deterministic Finite Automaton (DFA)

A quintuple M = (K,∑,δ,s,F) where 1) K is a finite set of states. 2) ∑ is an alphabet. 3) δ : K × ∑ → K is a transition function. 4) s is the initial state. 5) F is a finite set of final states.

Question: When is a string accepted by M?

Answer: A string w ∈ ∑* is accepted by M if there exists a state q ∈ F such that (s,w) ⌒* (q,ε).

Question: What is an automaton?

Answer: An automaton is a machine designed to respond to encoded instructions.

Question: What is the language accepted by M?

Answer: The language accepted by M, L(M) is the set of all strings accepted by M.

Configuration

Any element (q,w) ∈ K × ∑*.