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I am wondering this statement above [the title] is true or not.

Here is what I've already had : non-recursive means undecidable.

I've read this Are all infinite languages undecidable?

which says:

If a Language is undecidable(non-recursive), there must be some strings make the TM fail to halt.SO IT MUST HAVE INFINITE OF THEM WHICH MAKE THE TM FAILS TO HALT.

How could this prove my statement[title]? Can anyone explain it to me a bit more clearly?

Thanks

ps. sorry for the confusion. Yes TM means Turing machine. And too be clear My question is : Does ALL non-recursive languages are Infinite?

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geasssos
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  • You seem to have a number of different concepts here. Are you wondering if `non-recursive == infinite` or `non-recursive == undecidable` or `infinite == undecidable` or something else entirely? Also, don't use abbreviations that people are unlikely to understand (although I'm guessing "TM" means "Turing Machine", based on the questions domain, at least). – twalberg Nov 20 '13 at 17:09
  • sorry for the confusion. Yes TM means Turing machine. And too be clear My question is : Does ALL non-recursive languages are Infinite? @twalberg – geasssos Nov 20 '13 at 17:19
  • Consider a language with an alphabet consisting of a single character (called A), and a single production "P -> A". The language accepts a single input, namely A, and is definitely non-recursive, and definitely non-infinite. So, no, not all non-recursive languages are infinite... – twalberg Nov 20 '13 at 17:23
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    This question appears to be off-topic because it is about computability theory, which is more appropriate at cs.stackexchange.com. – templatetypedef Nov 20 '13 at 18:18

1 Answers1

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As a hint: prove all finite languages are regular. All regular languages are decidable. Taking the contrapositive of this statement gives you that all undecidable (non-recursive) languages are infinite.

Hope this helps!

templatetypedef
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