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I'm curious about a potential relation between the following two languages.

$L_1 := \{\langle M_1, M_2 \rangle : L(M_1) \cap L(M_2) \ne \emptyset \}$.

$L_2 := \{\langle M_1, M_2 \rangle : L(M_1) \ne L(M_2) \}$.

Is it true that $L_2$ mapping reduces to $L_1$?

I've had many failed attempts to construct a reduction and would appreciate any help.

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Here are three hints.

  • Show that $L_2$ is not computably enumerable.
  • Show that $L_1$ is computably enumerable.
  • If $L_2$ mapping reduces to $L_1$, what does it imply?
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  • $\begingroup$ Are you sure that $L_2$ is not computably enumerable? $\endgroup$ – CuriousKid7 Mar 20 at 0:24
  • $\begingroup$ Try enumerating the pairs of TMs in $L_2$. You will find you just cannot find a way. Now prove indeed you cannot. $\endgroup$ – Apass.Jack Mar 20 at 0:25
  • $\begingroup$ Of course, if you can show that $L_2$ is not computably enumerable, then there is no such reduction. But could you please prove this? I really don't see why this is true. $\endgroup$ – CuriousKid7 Mar 20 at 0:31
  • $\begingroup$ But that language is computably enumerable. It's recognized by the universal Turing machine. $\endgroup$ – CuriousKid7 Mar 20 at 0:51
  • $\begingroup$ We can reduce $\overline{HALT}= \{⟨M, w⟩\mid\text{ TM }M\text{does not halt on input } w\}$ to $L_2$. $\endgroup$ – Apass.Jack Mar 20 at 0:54

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