Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

TL;DR

This paper explores the theory of infinite time Turing machines and applies it to analyze the hierarchy of equivalence relations on the reals, extending classical Borel reducibility concepts into the transfinite computational realm.

Contribution

It introduces a notion of infinite time reducibility that generalizes Borel reducibility to the class Δ¹₂, advancing the understanding of equivalence relations on the reals.

Findings

Developed the basic theory of infinite time Turing machines

Established a framework for infinite time reducibility

Extended Borel reducibility concepts into the transfinite setting

Abstract

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.