deterministic finite automaton
A deterministic finite automaton (or DFA) is a deterministic automaton with a finite input alphabet and a finite number of states. It can be formally defined as a 5tuple $(S,\mathrm{\Sigma},\delta ,{q}_{0},F)$, where

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$S$ is a nonempty finite set^{} of states,

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$\mathrm{\Sigma}$ is the alphabet (defining what set of input strings the automaton operates on),

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$\delta :S\times \mathrm{\Sigma}\to S$ is the transition function,

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${q}_{0}\in S$ is the starting state, and

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$F\subseteq S$ is a set of final (or accepting states).
A DFA works exactly like a general automaton: operation^{} begins at ${q}_{0}$, and movement from state to state is governed by the transition function $\delta $. A word is accepted exactly when a final state is reached upon reading the last (rightmost) symbol of the word.
DFAs represent regular languages, and can be used to test whether any string in ${\mathrm{\Sigma}}^{*}$ is in the language^{} it represents. Consider the following regular language over the alphabet $\mathrm{\Sigma}:=\{\U0001d68a,\U0001d68b\}$ (represented by the regular expression^{} aa*b
):
$$  $\text{::=}$  $\text{a}A$  
$$  $\text{::=}$  $b\U0001d68aA$ 
This language can be represented by the DFA with the following state diagram^{}: