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We know set A is countable if A is finite or in a one-to-one mapping to natural numbers.

Suppose ALPH be an arbitrary finite alphabet.

I summarize my inference:

a) Each arbitrary Language on ALPH is Countable. (I think this is True)

b) the set of all language from ALPH is Countable.(I think this is False)

c) for Each arbitrary Language on ALPH we have a generative formal grammar. (I think this is False)

d) Each arbitrary Language on ALPH that can be generated by formal grammar, is recursive. (I think this is True)

anyone could help me, and maybe correct me?

  • https://stackoverflow.com/q/26201270/781723, https://math.stackexchange.com/q/957967/14578, https://math.stackexchange.com/q/908743/14578 – D.W. Jan 08 '21 at 07:18

1 Answers1

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Without loss of generality, we can assume that ALPH is merely the set {0,1}. (Any other finite language can of course be encoded using the set {0,1}). Assuming that by a language L that you intend some arbitrary subset of ALPH*, we can assume that L is an arbitrary subset of {0,1}*.

Let S = {0,1}*.

a) The set S is countable. Since L is a subset of S, L is countable.

b) The set of all languages over S then is the powerset of S, which can be put into 1-1 correspondence with the Real numbers. Hence, not countable.

c) I believe that this is false, agreeing with your supposition. However, it depends on your definition of a 'generative formal grammar'. If you allow for formal grammars where individual rules of the grammar are undecidable, and/or allow for infinite generation rules, this becomes less clear. For any particular definition for 'generative formal grammar', where the collection of 'generative formal grammars' is enumerable, then of course, the answer is false.

d) In general, I believe that the answer to this is false. If you restrict yourself to formal grammars corresponding to context-free languages, then of course, your answer is true. However, consider http://en.wikipedia.org/wiki/Post_correspondence_problem The problem is undecidable, yet the generation rules are pretty clear.

Bill Province
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  • Dear Bill, you means just 4 is false? –  Oct 05 '14 at 18:09
  • I mean that only with item 4 (d), do my conclusions contradict your conclusions. (You provide a conjecture that both b and c are false, which I agree with). The situation is slightly less clear for (c), because it depends on your definitions. I suspect that for any 'well-behaved' notion of a generative formal grammar, then the statement (c) is false, agreeing with your supposition. I should add however, that this cannot be proven without a definition, so as-written, (c) has to be stated as a conjecture. – Bill Province Oct 05 '14 at 19:28