Toward higher-order infinite time Turing machines: simulational $\Gamma$-machines

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 $\lambda = \gamma$ holds for any of those limit rules. Eventually, we provide a counter-example to show that the four first constraints alone are not sufficient.