Randomness and Degree Theory for Infinite Time Register Machines

TL;DR

This paper introduces a new concept of randomness for infinite time register machines (ITRMs), explores its implications for computability and degrees, and examines autoreducibility properties.

Contribution

It defines a notion of randomness for ITRMs, proves key theorems like an analogue of van Lambalgen's theorem, and analyzes the structure of ITRM-degrees.

Findings

Randomness implies non-autoreducibility for ITRMs.

Computability from mutually random reals implies computability.

An analogue of van Lambalgen's theorem holds for ITRMs.

Abstract

A concept of randomness for infinite time register machines (ITRMs) is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies computability and that an analogue of van Lambalgen's theorem holds. This is then applied to obtain results on the structure of ITRM-degrees. Finally, we consider autoreducibility for ITRMs and show that randomness implies non-autoreducibility.