The computational strengths of α-tape infinite time Turing machines

Abstract

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≥ω). Let Tα denote the machine model of tape length α (so Tω is just the ITTM model). Define that Tα is computationally stronger than Tβ (abbreviated Tα≻Tβ) precisely when Tα can compute all Tβ-computable functions f:2min(α,β)→2min(α,β), plus more. The following results are found: (1) Tω1≻Tω. (2) There are countable ordinals α such that Tα≻Tω, the smallest of which is precisely γ, the supremum of ordinals clockable by Tω. In fact, there is a hierarchy of countable Tαs of increasing strength corresponding to the transfinite (weak) Turing-jump operator ∇. (3) There is a countable ordinal μ such that neither Tμ⪰Tω1 nor Tμ⪯Tω1—that is, the models Tμ and Tω1 are computation-strength incommensurable (and the same holds if countable μ′>μ replaces μ). A similar fact holds for any larger uncountable device replacing Tω1. (4) Further observations are made about countable Tα.