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arXiv · 2012.08001

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

Philip Welch
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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.

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Philip Welch. 2020-12-14. Characterisations of Variant Transfinite Computational Models: Infinite Time Turing, Ordinal Time Turing, and Blum-Shub-Smale machines. https://arxiv.org/abs/2012.08001

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