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Can someone please help me solve this problem? Let $HP = \{ \langle M, w\rangle: \text{$M$ is a Turing machine that halts on input $w$}\}$ denote the halting problem, and consider the language:

$$L_2 = \{\langle M \rangle : \text{$M$ is a Turing machine and }\overline{HP} \leq L(M)\}$$

Choose one of the following. The language $L_2$ is:

  1. trivial.
  2. a non-trivial language in $\text{R}$.
  3. in $\text{RE}\setminus \text{R}$.
  4. in $\overline{RE}$.

Prove the determined answer. I think it is 4, not in $\text{RE}$, but I am not sure how to prove it.

Bader Abu Radi
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yehr
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1 Answers1

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The correct answer is 0. Note that for every TM $M$, it holds by definition that $L(M)\in \text{RE}$ as every machine recognizes its own language. Therefore, there are no machines with $\overline{HP}\leq L(M)$. Indeed, otherwise, we get that the complement of halting problem $\overline{HP}$ is in $\text{RE}$ as well. So the language $L_2$ must be empty.

Bader Abu Radi
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