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Definition: A finite automaton is one of the most primitive forms of a computer or algorithm. It possesses finitely many state (like memory states) and transitions between the states that are triggered by an event or symbol. Such automata (plural of automaton) can simply recognize sequences of events or can produce output. Naturally, there is a formal mathematical definition that one can find in almost any elementary textbook. Finite automata are also called finite state machines.
Example: The simple addition algorithm is a finite automaton. An event is a pair of digits, a state is the current carry, the output is the first digit of the sum of the carry and the two digits, the transition is the next carry.