Characterisations of Variant Transfinite Computational Models: Infinite Time Turing, Ordinal Time Turing, and Blum-Shub-Smale machines

TL;DR

This paper explores how modifications in transfinite computational models affect their capabilities, using advanced set theory and logic to classify halting times, tape length admissibility, and the universality of Blum-Shub-Smale machines.

Contribution

It introduces new classifications of halting times, characterizes admissible tape lengths, and establishes the universality of a class of transfinite Blum-Shub-Smale machines.

Findings

Classified halting times of ITTMs and OTM using admissibility theory.

Determined admissible tape lengths for transfinite machines.

Proved the universality of a Blum-Shub-Smale machine with a $ mop{Liminf}$ rule.

Abstract

We consider how changes in transfinite machine architecture can sometimes alter substantially their capabilities. We approach the subject by answering three open problems touching on: firstly differing halting time considerations for machines with multiple as opposed to single heads, secondly space requirements, and lastly limit rules. We: 1) use admissibility theory, $\Sigma_{2}$-codes and $\Pi_{3}$-reflection properties in the constructible hierarchy to classify the halting times of ITTMs with multiple independent heads; the same for Ordinal Turing Machines which have $\tmop{On}$ length tapes; 2) determine which admissible lengths of tapes for transfinite time machines with long tapes allow the machine to address each of their cells - a question raised by B. Rin; 3) characterise exactly the strength and behaviour of transfinitely acting Blum-Shub-Smale machines using a $\tmop{Liminf}$ rule on their registers - thereby establishing there is a universal such machine. This is in contradistinction to the machine using a `continuity' rule which fails to be universal.