Toward higher-order infinite time Turing machines: simulational $Γ$-machines

TL;DR

The paper develops a unified framework for generalized infinite time Turing machines by isolating five structural properties (including looping stability) that ensure a universal machine exists and yield a generalized equality $\Sigma_\Gamma = \mathrm{T}_\Gamma$ for definable limit-rule operators. Specializing to two-symbol rules, the classical $\Gamma_{\sup}$ and $\Gamma_{\inf}$ fit these conditions and reproduce key ITTM phenomena, with corollaries such as $\zeta_\Gamma = \eta_\Gamma$ and $\lambda_\Gamma = \gamma_\Gamma$. A central result shows that for any definable simulational, looping-stable operator, the supremum of writable/clockable ordinals align ($\Sigma_\Gamma = \mathrm{T}_\Gamma$), generalizing the classical ITTM theory and linking ordinal computation to constructible hierarchy levels. The paper also proves the necessity of looping stability by constructing a counterexample where the main equality fails without it. Finally, it sketches higher-order, multi-symbol extensions and highlights ongoing questions, including how the Speed-up Lemma fails to port and what this implies for end-extension behavior in generalized ITTMs.

Abstract

Infinite time Turing machines (ITTMs) have been introduced by Hamkins and Lewis in their seminal article arXiv:math/9808093. The strength of the model comes from a limit rule which allows the ITTM to compute through ordinal stages. This rule is simple to describe and another rule would lead to a different model of ordinal computation. The aim of this article is to define a collection of limit rules for which the models of infinite Turing machine they induce have nice properties, akin to that of the ITTM. Through the analysis of the universal ITTM and of its preponderant role in the study of the ITTM, we devise a set of four constraints. A limit rule satisfying those constraints yields a model of infinite machine for which we can define a universal machine. Adding a fifth constraint, we show that any limit rule definable in set theory which meets those constraints produces a model of infinite machine with the desired properties. Among those, the fact that the supremum of the writable ordinal matches that of the clockable ordinal. That is, with the usual notations, the equality $λ= γ$ holds for any of those limit rules. Eventually, we provide a counter-example to show that the four first constraints alone are not sufficient.

Figures (8)

• Figure 1: Greek letters associated to each definition.
• Figure 2: The snapshot $s_0$ repeats $\omega$ times and if the snapshot $s_1$ does not appear for the first time after this repetition it must have appeared before stage $\alpha_0$.
• Figure 3: The segment that spans between $\beta_0$ and $\beta_2$ repeats $\omega$ times and produces $x'$.
• Figure 4: By elementarity, there is $\delta_2' < \beta_2$ such that $C^m(\delta_2') = x'$
• Figure 5: Situation of the main constants of the operator $\Gamma$ if $\Sigma_\Gamma < \eta_\Gamma$.

Theorems & Definitions (140)

• Definition 1.1: hamkins_ittm
• Proposition 2.1: welch_main
• proof
• Definition 3.1: Additively closed ordinals
• Definition 3.2: Multiplicatively closed ordinals
• Definition 3.3: Ordinal word
• Example 3.4
• Definition 3.5: Stutter-free
• Definition 3.6: Suffixes and prefixes
• Definition 3.7: Operations on words