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I have just began with my course of complexity and computability and I need your experience to help me progress !!

Something is not clear for me:

Exercise

It is asked to determine if L1 is decidable or not, below the language is an answer given by a student. And I'm not agree with it...

The Rice theorem is given as follow :

Rice Theorem

I don't get why L1 is decidable since, for me, it looks like the Rice theorem could be immediately applied.

Did I understood very badly or am I right ? In the first case, why ?

Thanks in advance for your help !

Elia

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  • $\begingroup$ Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources! $\endgroup$ – dkaeae Apr 4 at 7:08
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Rice's theorem only applies for nontrivial properties, which are properties which are not always true or not always false. Your property is trivial, since it's always true.

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  • $\begingroup$ Thanks for your answer !! Is it trivial because L1 is actually equal to RE ? $\endgroup$ – Elia Dratwa Apr 4 at 0:25
  • $\begingroup$ The semantic property is trivial and so the language is recursive. $\endgroup$ – Yuval Filmus Apr 4 at 5:15
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    $\begingroup$ @EliaDratwa L1 can not be equal to RE: L1 is a language, RE is a set of languages. The issue here is that $L1 = \{\langle M\rangle\ |\ M \mbox{is a TM}\}$, so to decide L1 we only need to understand whether a word is a description of some TM, which is a simple "parsing" problem, so to speak. $\endgroup$ – chi Apr 4 at 14:25
  • $\begingroup$ Thanks a lot !! $\endgroup$ – Elia Dratwa Apr 6 at 14:31

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