Suppose an enumerator E can enumerate some language L in a finite amount of time (3 days, 3 hours, etc.). What properties does this language have?

My guess:

  • L is finite, so L is a context-free language.
  • Since L is a context-free language, L is also decidable
  • Since L is decidable, L is also Turing-recognizable and co-Turing recognizable

My Questions:

  • Is L also a regular language?
  • A theorem says that:

A language is Turing-recognizable iff some enumerator enumerates it

  • Is there anything more or less we can say about L, compared to a Turing-recognizable language?

If the language can be enumerated in finite time, then the language itself is finite. Almost all natural classes of languages include all finite languages: they're regular, context-free, decidable, recursively enumerable, co-RE, ...

I'm not sure what else you'd want to say about finite languages compared to RE languages. It's enough to say that the language is finite; for any property you might be interested in, it's usually very straightforward to prove that finite languages do or do not have that property.

  • $\begingroup$ Thank you! So the very least you could say is that it's decidable (and therefore RE, and co-RE)? $\endgroup$ – user3400223 Dec 13 '16 at 20:53

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