I have found the following task: Show that K (the halting set) is m-reducable to the zero functions $$ K \leq_M\{x | f_x = 0\}$$
We need to find a total-computable function h, such that: $$ x \in K \iff h(x) \in \{x | f_x = 0\} $$ The i have found this solution:
h(x,y) = t (if f_x (x)↓}
h(x,y) = ↑ (else)
But shouldn't h be a total-computable function? And h is not-total ? And why does it have two arguments? Any tips? Am I understanding things wrong?
page 5: http://www.cs.sjtu.edu.cn/~gao-xf/computability/Document/10-Reducibility.pdf
Your understanding is correct; the slides are somewhat confusingly written.
The function $f_{\bf 0}$ is not the desired $m$-reduction. Indeed it can't possibly be since the arity is wrong: an $m$-reduction is unary, but $f_{\bf 0}$ is binary. Instead, $f_{\bf 0}$ is an oddly-packaged description of the outputs of the intended $m$-reduction of $K$ to $\{x: \phi_x\simeq{\bf 0}\}$.
The point is that for each $x$, the function $$\eta_x: y\mapsto f_{\bf 0}(x,y)$$ is a partial computable function with the property that $\eta_x\simeq {\bf 0}$ iff $x\in K$. (I'm writing "$\simeq$," rather than "$=$," for equality of possibly-partial functions.)
The idea is then that our $h$ should be a function sending $x$ to an index for the partial computable function $\eta_x$. That is:
The existence of such an $h$ is then justified by the s-m-n theorem, applied to the partial computable binary function $f_{\bf 0}$. So when the author more-or-less says "$f_{\bf 0}(x,y)$ demonstrates $K\le_m\{x: \phi_x\simeq {\bf 0}\}$," what they mean is that we can extract an appropriate reduction from $f_{\bf 0}$, not that $f_{\bf 0}$ is itself such a reduction.