Consider the reduction $A_{TM} \le_m \overline{E}_{TM}$, where
$$A_{TM} = \{\langle M, w \rangle \mid \text{TM $M$ accepts $w$}\}\text{, and}$$
$$\overline{E}_{TM} = \{\langle M \rangle \mid \text{TM $M$ accepts some string}\}$$
Then define the mapping reduction, $f(\langle M, w \rangle)$ as follows:
Construct a new TM $M^\prime$ that on input $x$—
- rejects if $x \ne w$;
- otherwise, runs $M(w)$ and accepts if it accepts.
Finally, it outputs $\langle M^\prime \rangle$. This construction ensures that $\langle M, w \rangle \in A_{TM} \Leftrightarrow \langle M^\prime \rangle \in \overline{E}_{TM}$.
Question. The idea makes sense, except that I am having a hard time wrapping my head around infinite loops and the computability of mapping reductions. In particular, what ensures that we can indeed write $\langle M^\prime \rangle$ even if $M$ does not halt on $w$? It seems like the answer should be obvious — that there is some generic way to encode $\langle M^\prime \rangle$, but I don't know how.
Secondly, is it correct to say that $f$ is computable because it is fully determined by its inputs?
