Undecidability of the halting problem

Question

One can prove by reduction from the special halting problem $H_S$ the undecidability of the (general) halting problem $H$. Is the converse also possible? That is, is it possible to prove the undecidability of the special halting problem $H_S$ by reduction from the (general) halting problem $H$?

The language $H_S$ of the special halting problem:

$H_S := \{w: w \mbox{ encodes a Turing machine }M_w \mbox{ and } M_w \mbox{ accepts } w \}$

The language $H$ of the (general) halting problem:

$H:=\{\langle w,x\rangle\;:\;w \mbox{ encodes a Turing machine }M_w \mbox{ and } M_w \mbox{ accepts } x\}$

In both definitions $w,x\in \Sigma^*$ for some alphabet $\Sigma$.

Answer 1

Here is how to reduce the general problem to the special one. Given a machine $w$ and input $x$, apply the s-m-n theorem to obtain a machine $w'$ which, regardless of its input, simulates the execution of machine $w$ with input $x$. Then $w' \in H_S$ if and only if $\langle w,x\rangle \in H$.

The opposite reduction is trivial: $w \in H_S$ if and only if $\langle w,w\rangle \in H$.

Answer 2

In short, yes. In the special halting problem, you want to determine whether an arbitrary Turing machine with null input will halt or not, while with the general halting problem you want to know whether an arbitrary Turing machine with arbitrary input will halt or not. Clearly, since the special problem is a particular case of the general one, proving the general is sufficient to prove the special.

However, it's a lot harder to prove the general halting problem than the special one, hence why you tend to prove the special one, and from that prove the general one.