Computable Component-wise Reducibility | Egor Ianovski

We consider equivalence relations and preorders complete for various levels of the arithmetical hierarchy under computable, component-wise reducibility. We show that implication in first order logic is a complete preorder for Σ1, the ≤m relation on EXPTIME sets for Σ2 and the embeddability of computable subgroups of (Q,+) for Σ3. In all cases, the symmetric fragment of the preorder is complete for equivalence relations on the same level. We present a characterisation of Π1 equivalence relations which allows us to establish that equality of polynomial time functions and inclusion of polynomial time sets are complete for Π1 equivalence relations and preorders respectively. We also show that this is the limit of the enquiry: for n ≥ 2 there are no Πn nor ∆n-complete equivalence relations.