Complexity of equivalence relations and preorders from computability theory
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 $Π_1$-complete equivalence relation, but no $Πk$-complete for $k \ge 2$. 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 $Σ4$.