Computable Component-wise Reducibility

TL;DR

This paper explores the complexity of various equivalence relations and preorders across the arithmetical hierarchy, establishing completeness results for several logical and computational structures.

Contribution

It introduces new completeness results for equivalence relations and preorders at different levels of the arithmetical hierarchy, including characterizations and limitations.

Findings

Implication in first order logic is a complete preorder for .

The  level includes ^P_m relation on EXPTIME sets.

Equality of polynomial time functions is -complete.

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.