Computable Component-wise Reducibility

Abstract

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 $\SI 1$, the $\le^P_m$ relation on EXPTIME sets for $\SI 2$ and the embeddability of computable subgroups of $(\QQ,+)$ for $\SI 3$. In all cases, the symmetric fragment of the preorder is complete for equivalence relations on the same level. We present a characterisation of $\PI 1$ equivalence relations which allows us to establish that equality of polynomial time functions and inclusion of polynomial time sets are complete for $\PI 1$ equivalence relations and preorders respectively. We also show that this is the limit of the enquiry: for $n\geq 2$ there are no $\PI n$ nor $\DE n$-complete equivalence relations.