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Automata Theory (Definition)

The study of abstract machines (automata) and the problems they can solve.
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The statement of the theorem

Let Σ\Sigma be a finite alphabet. A Deterministic Finite Automaton (DFA) is a 5-tuple M=⟨Q,Σ,δ,q0,F⟩M = \langle Q, \Sigma, \delta, q_0, F \rangle, where:\\ \n1. QQ is a finite set of states.\\ 2. Σ\Sigma is the input alphabet.\\ 3. δ:Q×Σ→Q\delta: Q \times \Sigma \to Q is the transition function.\\ 4. q0∈Qq_0 \in Q is the initial state.\\ 5. F⊆QF \subseteq Q is the set of accepting (final) states.\\ \nFor any input string w=w1w2…wn∈Σ∗w = w_1 w_2 \dots w_n \in \Sigma^*, the extended transition function δ∗:Q×Σ∗→Q\delta^*: Q \times \Sigma^* \to Q is defined recursively:\\ \nδ∗(q,w1w2…wn)=δ∗((δ∗(q,w1w2…wn−1)),wn)\delta^*(q, w_1 w_2 \dots w_n) = \delta^*((\delta^*(q, w_1 w_2 \dots w_{n-1})), w_n) \\nwith the base case δ∗(q,ϵ)=q\delta^*(q, \epsilon) = q. \\nThe language recognized by MM, denoted L(M)L(M), is the set of all strings ww that lead the automaton from the initial state q0q_0 to an accepting state f∈Ff \in F: \\nL(M)={w∈Σ∗∣δ∗(q0,w)∈F}L(M) = \{ w \in \Sigma^* \mid \delta^*(q_0, w) \in F \} \\nAutomata Theory studies the relationship between the class of formal languages (e.g., regular languages, context-free languages) and the minimal computational models (automata) required to recognize them, formalized by theorems such as the Pumping Lemma and the Chomsky Hierarchy.