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 Π1-complete equivalence relation, but
no Πk-complete for k≥2.
We show that Σk preorders arising naturally in the above-mentioned
areas are Σk-complete. This includes polynomial time m-reducibility
on exponential time sets, which is Σ2, almost inclusion on r.e.\ sets,
which is Σ3, and Turing reducibility on r.e.\ sets, which is Σ4.Comment: To appear in J. Symb. Logi