 Solution

Membership is immediate from the supplied characterization of ${\Pi }_2$. Define the partial computable functionThenso $\mathbf{Tot}\in {\Pi }_2$.

Now let $P\in {\Pi }_2$. There is a partial computable $f$ such thatFor each $w$, define a unary partial functionOperationally, its program simulates $f(w,x)$ and returns $x$ if that computation halts. By the S-m-n theorem, a total computable map $h$ produces an index $h(w)$ for $g_w$. ThereforeThus every Pi-2 set many-one reduces to $\mathbf{Tot}$. Combined with membership, this proves that the totality problem is ${\Pi }_2$-complete.