The computational strengths of α-tape infinite time Turing machines

In [7], open questions are raised regarding the computational strengths of so-called ∞-α-Turing machines, a family of models of computation resembling the infinite-time Turing machine (ITTM) model of [2], except with α  -length tape (for α≥ωα≥ω). Let TαTα denote the machine model of tape length α   (so TωTω is just the ITTM model). Define that TαTα is computationally stronger   than TβTβ (abbreviated Tα≻TβTα≻Tβ) precisely when TαTα can compute all TβTβ-computable functions f:2min(α,β)→2min(α,β), plus more. The following results are found: (1) Tω1≻TωTω1≻Tω. (2) There are countable ordinals α   such that Tα≻TωTα≻Tω, the smallest of which is precisely γ  , the supremum of ordinals clockable by TωTω. In fact, there is a hierarchy of countable TαTαs of increasing strength corresponding to the transfinite (weak) Turing-jump operator ∇. (3) There is a countable ordinal μ   such that neither Tμ⪰Tω1Tμ⪰Tω1 nor Tμ⪯Tω1Tμ⪯Tω1—that is, the models TμTμ and Tω1Tω1 are computation-strength incommensurable (and the same holds if countable μ′>μμ′>μ replaces μ  ). A similar fact holds for any larger uncountable device replacing Tω1Tω1. (4) Further observations are made about countable TαTα.