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

Article type: Research Article

Authors: Welch, Philip^{; *}

Affiliations: School of Mathematics, University of Bristol, England. [email protected]

Correspondence: [*] Corresponding author. [email protected].

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, Σ2-codes and Π3-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.

Keywords: Turing Machine, computability, Blum–Shub–Smale, generalized recursion, admissible set

DOI: 10.3233/COM-200301

Journal: Computability, vol. 10, no. 2, pp. 159-180, 2021