Lets say I have just used the pumping lemma to prove a certain L language is not CFL.

If it is not CFL can I use that as a proof that it is Decidable? Or is this not enouph and I still have to provide a Turing Machine that accepts or rejects L?


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    $\begingroup$ If something costs more than one dollar, can I use that as a proof that it costs less than ten dollars? $\endgroup$ – David Richerby Jul 1 '16 at 14:11
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    $\begingroup$ Why would you think that you can assume it is decidable in that case? We're more likely to be able to help you if you edit the question to explain your reasoning and your thought process. $\endgroup$ – D.W. Jul 1 '16 at 14:16
  • $\begingroup$ My reasoning was that Decidability is the next "step up" from CFL, so to say. I can see now that it if is not a CFL, it could be decidable, recognizable or even non-recognizable. $\endgroup$ – Ricardo Ferreira da Silva Jul 5 '16 at 20:16

No undecidable language is context free, so after proving non-CFL, it could still be decidable or not. You would need to prove that separately.


Every undecidable language is not context-free, since all context-free languages are decidable. Since there exist undecidable languages, we conclude that there exist non-context-free languages which are undecidable.


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