If one can decide the set membership problem for a language by means of a computer program, does it state that the language is regular?

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So, all languages for which if I can decide the set membership problem by means of computer programs, is it possible, or correct to say that all such languages are regular? If not, why?

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Depends what you mean by a computer program. If you mean a deterministic finite automaton (DFA), then yes.

Kleene's Theorem. A language is regular if and only if it is recognisable by some DFA.