Lower bounds on β ( α ) and other properties of α-register machines

Abstract

This paper extends our paper (In Revolutions and Revelations in Computability. CiE 2022 (2022)) for the conference “Computability in Europe” 2022. After Infinite Time Turing Machines (ITTM) were introduced in Hamkins and Lewis (Journal of Symbolic Logic 65 (2000)), a number of machine models of computability have been generalized to the transfinite. While for some of these models the computational strength has been successfully determined, there are still several white spots on the map of transfinite computability. In this paper, we contribute to the understanding of the computational strength of transfinite machine models by (i) proving lower bounds on the computational strength of α-Infinite Time Register Machines (α-ITRMs) for certain values of α, refuting a conjecture about their strength made in (Annals of Pure and Applied Logic 173 (2022)), (ii) showing that the computational strength of cardinal-recognizing ITRMs is equal to that of ITRMs and (iii) showing that non-solvability of the bounded halting problem, existence of a universal machine and an increase of computational power by allowing machines to recognize cardinals are equivalent for α-ITRMs for all relevant values of α. Finally, we give some results indicating how the picture changes when the use of parameters is dropped or restricted.