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

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.