Q: Is this function computable? Given the function
$$  f(x) =
\begin{cases}
1,  & \text{if $x = W(M)$ is the encoding of a turing machine $M$ and $M$ halts if the empty word is entered } \\
undefined, & \text{otherwise}
\end{cases}$$
is it computable? $f$ is defined on $\Bbb N$. I think one of the problem here is to decide whether some $x$ is indeed the encoding of a turing machine or not. I don't see how one should be able to decide something like that. 
On the other hand, if we know that something is the encoding of a turing machine, it should be pretty easy to decide whether the turing machine halts if the empty word is entered, although I'm not sure how to formalise this.
The function being undefined isn't a problem, this can be achieved by defining a loop.
