Complement of $\{w\#x \mid w,x \in \Sigma^*, T(M_w) \neq \{x\}\}$

Question

This is one of my homework assignment questions, that are quite difficult for me. The question states:

Show that $L$ is not semi-decidable where $L = \{w\#x \mid w,x \in \Sigma^*, T(M_w)\neq \{x\}\}$

So I just need a better understanding of $L$ in order to solve this homework. My question now is:

Is the complement of $L$ the language $\overline{L} = \{w\#x \mid w,x \in \Sigma^*, T(M_w) = \{x\}\}$?

If yes, I think it's possible to show that $\overline{H_0} \leq \overline{L}$, but that's the second step that I would like to figure out by myself.

Answer 1

The complement of a language $L$ over an alphabet $\Delta$ consists of all words over $\Delta$ which do not belong to $L$. In your case, $\Delta = \Sigma \cup \{\#\}$. So a string $z$ is in $\overline{L}$ if it is not of the form $w\#x$, where $w,x \in \Sigma^*$ and $T(M_w) \neq \{x\}$. This can happen in two ways:

• $z$ is not of the form $\Sigma^* \# \Sigma^*$.
• $z = w\#x$, where $w,x \in \Sigma^*$ and $T(M_w) = \{x\}$.