Towards a Church-Turing-Thesis for Infinitary Computations

TL;DR

This paper explores the concept of transfinite computability, proposing a canonical model called Idealized Agent Machines, and demonstrates its equivalence to existing models like Ordinal Turing Machines, aiming to extend the Church-Turing thesis into the transfinite realm.

Contribution

It introduces Idealized Agent Machines as a new canonical model for transfinite computation and proves their equivalence to Ordinal Turing Machines, advancing the understanding of infinitary computability.

Findings

Idealized Agent Machines are equivalent to Ordinal Turing Machines

Proposes a transfinite analogue of the Church-Turing thesis

Establishes a canonical model for transfinite computability

Abstract

We consider the question whether there is an infinitary analogue of the Church-Turing-thesis. To this end, we argue that there is an intuitive notion of transfinite computability and build a canonical model, called Idealized Agent Machines ($IAM$s) of this which will turn out to be equivalent in strength to the Ordinal Turing Machines defined by P. Koepke.