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Formal definition

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The statement of the theorem

Automaton An automaton can be represented formally by a quintuple M=⟨Σ,Γ,Q,δ,λ⟩M=\langle \Sigma ,\Gamma ,Q,\delta ,\lambda \rangle , where: - Σ\Sigma is a finite set of symbols, called the input alphabet of the automaton, - Γ\Gamma is another finite set of symbols, called the output alphabet of the automaton, - QQ is a set of states, - δ\delta is the next-state function or transition function δ:Q×Σ→Q\delta :Q\times \Sigma \to Q mapping state-input pairs to successor states, - λ\lambda is the next-output function λ:Q×Σ→Γ\lambda :Q\times \Sigma \to \Gamma mapping state-input pairs to outputs. If QQ is finite, then MM is a finite automaton. Input word An automaton reads a finite string of symbols a1a2...ana_{1}a_{2}...a_{n} , where ai∈Σa_{i}\in \Sigma , which is called an input word. The set of all words is denoted by Σ∗\Sigma ^{*} . Run A sequence of states q0,q1,...,qnq_{0},q_{1},...,q_{n} , where qi∈Qq_{i}\in Q such that qi=δ(qi−1,ai)q_{i}=\delta (q_{i-1},a_{i}) for 0<i≤n0<i\leq n , is a run of the automaton on an input a1a2...an∈Σ∗a_{1}a_{2}...a_{n}\in \Sigma ^{*} starting from state q0q_{0} . In other words, at first the automaton is at the start state q0q_{0} , and receives input a1a_{1} . For a1a_{1} and every following aia_{i} in the input string, the automaton picks the next state qiq_{i} according to the transition function δ(qi−1,ai)\delta (q_{i-1},a_{i}) , until the last symbol ana_{n} has been read, leaving the machine in the final state of the run, qnq_{n} . Similarly, at each step, the automaton emits an output symbol according to the output function λ(qi−1,ai)\lambda (q_{i-1},a_{i}) . The transition function δ\delta is extended inductively into δ‾:Q×Σ∗→Q{\overline {\delta }}:Q\times \Sigma ^{*}\to Q to describe the machine's behavior when fed whole input words. For the empty string ε\varepsilon , δ‾(q,ε)=q{\overline {\delta }}(q,\varepsilon )=q for all states qq , and for strings wawa where aa is the last symbol and ww is the (possibly empty) rest of the string, δ‾(q,wa)=δ(δ‾(q,w),a){\overline {\delta }}(q,wa)=\delta ({\overline {\delta }}(q,w),a) . The output function λ\lambda may be extended similarly into λ‾(q,w){\overline {\lambda }}(q,w) , which gives the complete output of the machine when run on word ww from state qq .Acceptor In order to study an automaton with the theory of formal languages, an automaton may be considered as an acceptor, replacing the output alphabet and function Γ\Gamma and λ\lambda with - q0∈Qq_{0}\in Q , a designated start state, and - FF , a set of states of QQ (i.e. F⊆QF\subseteq Q ) called accept states. This allows the following to be defined: Accepting word A word w=a1a2...an∈Σ∗w=a_{1}a_{2}...a_{n}\in \Sigma ^{*} is an accepting word for the automaton if δ‾(q0,w)∈F{\overline {\delta }}(q_{0},w)\in F , that is, if after consuming the whole string ww the machine is in an accept state. Recognized language The language L⊆Σ∗L\subseteq \Sigma ^{*} recognized by an automaton is the set of all the words that are accepted by the automaton, L={w∈Σ∗ ∣ δ‾(q0,w)∈F}L=\{w\in \Sigma ^{*}\ |\ {\overline {\delta }}(q_{0},w)\in F\} . Recognizable languages The recognizable languages are the set of languages that are recognized by some automaton. For finite automata the recognizable languages are regular languages. For different types of automata, the recognizable languages are different. - ^Cite error: The named reference Arbib 1969 was invoked but never defined (see the help page). - ^Cite error: The named reference structure theory was invoked but never defined (see the help page). - ^Moore, Cristopher (2019-07-31). "Automata, languages, and grammars". arXiv:1907.12713 [cs.CC].