  We study the relative complexity of equivalence relations and preorders from
computability theory and complexity theory. Given binary relations $R, S$, a
componentwise reducibility is defined by $ R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. $ Here $f$ is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and $f$ must be
computable. We show that there is a $\Pi_1$-complete equivalence relation, but
no $\Pi k$-complete for $k \ge 2$.
  We show that $\Sigma k$ preorders arising naturally in the above-mentioned
areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility
on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets,
which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma
4$.
