Primitive recursive equivalence relations and their primitive recursive complexity

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 : ω → ω 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 Peq, the degree structure generated by primitive recursive reducibility on primitive recursive equivalence relations with infinitely many equivalence classes. In contrast with all other known degree structures on equivalence relations, we show that 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 Peq, proving, e.g., that the structure is neither rigid nor homogeneous.