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We explore the borderline between decidability and
undecidability of the following question: “Let C be a
class of codes. Given a machine ${\mathfrak{M}}$
of type X, is it decidable
whether the language $L({{\mathfrak{M}}})$
lies in C or not?” for
codes in general, ω-codes, codes of finite and bounded
deciphering delay, prefix, suffix and bi(pre)fix codes, and for
finite automata equipped with different versions of push-down stores
and counters.
