A finite-state machine processes inputs using a finite set of automaton states and input-dependent automaton transitions. An automaton state records everything the machine remembers; an automaton transition says how that memory changes on the next symbol. A finite-state machine may also emit outputs. Here the emphasis is on recognizing a formal language: start in a designated automaton state, read a word from an alphabet, and accept exactly when the final automaton state is designated accepting.
Finite memory does not impose a maximum input length. A finite-state machine can read arbitrarily long words, revisiting its automaton states indefinitely. The restriction is that the part already read must be compressed into one of finitely many summaries. For divisibility by three, a word of binary digits can describe an enormous integer, yet its congruence class modulo three is all that the machine needs to retain. Two already-read words giving the same automaton state behave identically under every subsequent continuation in a DFA.
This makes finite-state machines useful whenever past details can be forgotten safely: checking whether a sequence of commands follows an allowed order, or recognizing a pattern with a regular expression. The beautiful design question is which distinctions about the past matter to the future. DFA minimization makes that question precise by merging indistinguishable states of a deterministic finite automaton. Both DFAs and NFAs recognize exactly the regular languages.
New to topics? Read the docs here!