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I have a problem giving "intuitive" explanation to turing-unrecognizable languages.

We can prove that, say, ${\overline{A_{TM}}}$ is not turing-recognizable, because that would make ${{A_{TM}}}$ decidable, but what would the $TM$ do if it doesn't accept, reject of loop?

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You might be a little confused by a definition somewhere. A TM $M$ is a recognizer for a language $L$ if it

  • Accepts every string in $L$
  • Either rejects or loops on every string not in $L$

Now let's think about the language ${\overline{A_{TM}}}$. An input $\langle N,x\rangle$ is in ${\overline{A_{TM}}}$ if either:

  • $N$ rejects on $x$.
  • $N$ loops on $x$

So any recognizer will need to figure out (in finite time) that $N$ is going to loop on some input, which is just the halting problem.

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  • $\begingroup$ Thanks for connecting it to the halting problem. From there I stumbled on this resource and it sort of brought things to order cgl.uwaterloo.ca/~csk/halt. $\endgroup$
    – alex440
    Commented May 9, 2014 at 17:48

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