Characterisations of variant transfinite computational models: Infinite time Turing, ordinal time Turing, and Blum–Shub–Smale machines

Author:

Welch Philip1

Affiliation:

1. School of Mathematics, University of Bristol, England. p.welch@bristol.ac.uk

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, Σ 2 -codes and Π 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 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 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.

Publisher

IOS Press

Subject

Artificial Intelligence,Computational Theory and Mathematics,Computer Science Applications,Theoretical Computer Science

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1. Computation and Hypercomputation;Mathematics;2022-03-20

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