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automata theory, finite state machines, quantum mechanics, category theory, software development.
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Regular expressions are an algebraic notation to describe regular languages. They can be converted into minimal deterministic finite automata with the elegant Brzozowski algorithm.
Automata can be generalized by adding weights to transitions. This way they can compute more complicated functions than just predicates.
Why are automata so interesting? Automata are so simple that they fit in many places and situations.