Algorithmic Randomness for Infinite Time Register Machines

TL;DR

This paper introduces and analyzes a notion of randomness for infinite time register machines, demonstrating its properties and parallels to classical randomness concepts like Martin-Löf-randomness.

Contribution

It defines a new randomness concept for ITRMs and proves key properties including computability implications and an analogue of van Lambalgen's theorem.

Findings

Computability from mutually random reals implies computability.

An analogue of van Lambalgen's theorem holds for this randomness.

The new randomness notion resembles Martin-Löf-randomness.

Abstract

A concept of randomness for infinite time register machines (ITRMs), resembling Martin-L\"of-randomness, 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.