The Complexity of Index Sets of Classes of computably Enumerable Equivalence Relations

Let ďc be computable reducibility on ceers. We show that for every computably enumerable equivalence relation (or ceer) R with infinitely many equivalence classes, the index sets ti : Ri ďc Ru (with R non-universal), ti : Ri ěc Ru, and ti : Ri ”c Ru are Σ3 complete, whereas in case R has only finitely many equivalence classes, we have that ti : Ri ďc Ru is Π2 complete, and ti : Ri ěc Ru (with R having at least two distinct equivalence classes) is Σ2 complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is Π4 complete. Finally, we prove that the 1-reducibility pre-ordering on c.e. sets is a Σ 0 3 complete pre-ordering relation, a fact that is used to show that the pre-ordering relation ďc on ceers is a Σ3 complete pre-ordering relation.