Punctual equivalence relations and their (punctual) complexity

TL;DR

This paper investigates the structure of punctual equivalence relations under primitive recursive reducibility, revealing a dense lattice structure with complex, non-rigid properties unlike other known equivalence relation degree structures.

Contribution

It introduces and analyzes the degree structure generated by primitive recursive reducibility on punctual equivalence relations, showing its rich lattice structure and intricate properties.

Findings

The degree structure is a dense distributive lattice.

It is neither rigid nor homogeneous.

It exhibits more complexity than other known equivalence relation structures.

Abstract

The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations $R$ and $S$ on natural numbers, $R$ is computably reducible to $S$ if there is a computable function $f \colon \omega \to \omega$ that induces an injective map from $R$-equivalence classes to $S$-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the polynomial-time reducibility. In this work, we explore $\mathbf{Peq}$, the degree structure generated by primitive recursive reducibility on punctual equivalence relations (i.e., primitive recursive equivalence relations with domain $\omega$). In contrast with all other known degree structures on equivalence relations, we show that $\mathbf{Peq}$ has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of $\mathbf{Peq}$, proving, e.g., that the structure is neither rigid nor homogeneous.