I'm studying for my finals and I came across this question from past exams:

Is the following language decidable? $$ L = \{ \langle M_1,M_2,w \rangle \mid w \in L(M_1) \setminus L(M_2) \}. $$

How can I be sure of my answer? Is there any way to know for sure?

  • 2
    $\begingroup$ The way we know for sure in mathematics is by proving things. $\endgroup$ Jan 15 at 11:22

Intuitively, $L$ is not decidable as given $\langle M_1, M_2, w\rangle$ we need to decide whether

  1. $w\in L(M_1)$, and
  2. $w\in \overline{L(M_2)}$.

The first condition smells like deciding the language $A_{TM} = \{ \langle M, w \rangle: \text{ $M$ accepts $w$}\}$, and the second condition smells like deciding $\overline{A_{TM}}$. To make sure that the intuition got it right, you need to prove formally undecidability for $L$, here is a hint.

Hint: show that there is a reduction from some undecidable language to $L$. It is sufficient to capture the hardness of one the above conditions. Thus, to make things simpler, think how you can fix one of the machines $M_1$ or $M_2$ in the reduction's output, so that one of the conditions holds trivially. I leave the rest to you.


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