The universality of polynomial time Turing equivalence

TL;DR

This paper demonstrates that polynomial time Turing equivalence and similar relations are universal countable Borel equivalence relations, and explores ultrafilters related to Martin's ultrafilter on Turing degrees.

Contribution

It establishes the universality of polynomial time Turing equivalence within the framework of Borel equivalence relations and discusses related ultrafilters.

Findings

Polynomial time Turing equivalence is a universal countable Borel equivalence relation.

Many other equivalence relations in computational complexity are also universal.

Connections between ultrafilters on invariant Borel sets and Martin's ultrafilter are explored.

Abstract

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.