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I thought to reduce from the halting problem to conclude undecidability, yet I don't know how to do it. Perhaps the problem reduces to other decidable problem, and thus it is also decidable?

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  • $\begingroup$ It has been a few years since my theory course but if I recall right, that is undecidable. Look into Rice's theorem, it may be useful for this. $\endgroup$
    – Hank Igoe
    Jan 9, 2021 at 10:37
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    $\begingroup$ I suggest you keep trying. $\endgroup$ Jan 9, 2021 at 10:45
  • $\begingroup$ @HankIgoe I did it differently as according to you you would by Rice's theorem? $\endgroup$
    – Paradojin
    Jan 9, 2021 at 21:47
  • $\begingroup$ @Paradojin Here is a demonstration of how it can be done with Rice's theorem, if you're interested: cs.stackexchange.com/questions/19876/… $\endgroup$ Jan 10, 2021 at 10:03

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As you suggested, think how to reduce from the halting problem $Halt_{TM} = \{ \langle M, w\rangle: \text{$M$ halts on $w$} \}$. On input $\langle M, w\rangle$, the reduction should output a pair of TMs $\langle K_1, K_2\rangle$; such that $M$ halts on $w$ iff $L(K_1) = L(K_2)$.

This is somehow a basic reduction that can be done by standard tricks, so it would be a nice exercise to solve it alone. Here is a hint.

Hint: you can simplify things by fixing one of the machines $K_1$ or $K_2$ to be a constant machine that recognizes some fixed language. Then, think how to define the other machine depending on the reduction's input $\langle M, w\rangle$.

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  • $\begingroup$ thanks this helped me a lot! $\endgroup$
    – Paradojin
    Jan 9, 2021 at 17:19
  • $\begingroup$ You're welcome. I'm glad it did help. $\endgroup$ Jan 9, 2021 at 17:20

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