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Let

$\qquad \mathrm{DISJOINT} = \{ \langle M_1,M_2 \rangle : M_1, M_2 \text{ are TMs and } L(M_1) \cap L(M_2) = \emptyset\}$.

How do I know if this language is decidable or not? And How do I prove my answer?

Raphael
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Altaïr
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2 Answers2

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Assuming you have a decider $R$ for DISJOINT, you could use this to make a decider D for $E_\text{TM} = \{\langle M\rangle\mid L(M)=\emptyset\}$ as follows:

D(<M>) =
   return R(<M>, <A>)

where $A$ was a TM, selected in such a way that $\langle M\rangle\in E_\text{TM}$ if and only if $(\langle M\rangle, \langle A\rangle)\in\text{ DISJOINT}$. All that's left for you is to find the $A$ and show that it satisfies the needed conditions. (There are a couple of ways to make this choice.)

David Richerby
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Rick Decker
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Hint: If DISJOINT were decidable then even the special case in which $M_1$ is some fixed machine which accepts all inputs is decidable. This is the language $\{ \langle M \rangle : L(M) = \emptyset \}$, which you might be more familiar with.

Yuval Filmus
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  • I am confused. I don't understand. I think it's undecidable. Are you suggesting to Reduce $E_{TM}$ to $DISJOINT$ ? – Altaïr Nov 20 '14 at 18:51
  • Yes, that's another way of putting it. – Yuval Filmus Nov 20 '14 at 19:08
  • Are we gonna assume that DISJOINT is decidable and use that assumption to show that $E_{TM}$ IS decidable ? – Altaïr Nov 20 '14 at 19:55
  • Yes, that's the idea. – Yuval Filmus Nov 20 '14 at 19:55
  • So we assume we have a TM R that decides DISJOINT then we use R to construct TM S that decides $E_{TM}$ then we use R as subroutine to construct S. I just don't know how to write the algorithm. – Altaïr Nov 20 '14 at 20:01
  • It's gonna be something like proving $HALT_{TM}$ in Sipser's book. I just can't figure out what exactly to write – Altaïr Nov 20 '14 at 20:02
  • I've given you enough hints. I suggest you spend a few hours trying to completing the proof, and if you're still stuck, you can ask a new question about it. – Yuval Filmus Nov 20 '14 at 20:02
  • @RickDecker I can't find a reduction. Can you give me more hints ? I am trying to use R that decides DISJOINT as subroutine to construct TM S that decides $E_{TM}$ and write algorithm.(reduction) – Altaïr Nov 20 '14 at 20:12
  • See my expansion on the hints you've been given. The missing parts are up to you to fill in. – Rick Decker Nov 20 '14 at 20:31
  • @Altaïr This is a discussion you should have with a fellow student and/or teacher in front of board, on on some website. (As opposed to what Yuval advises, please bring your attempts and specific questions back to this question by editing it.) – Raphael Nov 20 '14 at 20:43