Infinite time decidable equivalence relation theory

TL;DR

This paper develops an infinite time computable framework for equivalence relations, extending classical Borel theory to include more complex relations beyond Borel or analytic, and explores their properties and reducibility.

Contribution

It introduces an infinite time analog of Borel equivalence relations and countable Borel relations, expanding the scope of the classical theory to more complex, beyond-Borel equivalence relations.

Findings

Many classical properties carry over to the infinite time setting

The theory includes equivalence relations beyond Borel and analytic classes

Results from Borel reducibility theory apply to the infinite time computable context

Abstract

We introduce an analog of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely Delta_1^2 reductions, and hence to the infinite time computable reductions.