Measuring the complexity of reductions between equivalence relations

TL;DR

This paper introduces degree spectra of reducibility and bi-reducibility to measure the complexity of equivalence relation reductions, demonstrating their rich structural properties and realizability.

Contribution

It generalizes computable reducibility by defining spectra that capture the complexity of reductions, showing they can have diverse and intricate structures.

Findings

Any upward closed collection of Turing degrees with a countable basis can be realized as a spectrum.

There exist spectra of computably enumerable equivalence relations with no countable basis.

Some spectra are downward dense and have no basis.

Abstract

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis.