Taming Koepke's Zoo II: Register Machines

Abstract

We study the computational strength of resetting $\alpha$-register machines, a model of transfinite computability introduced by P. Koepke in \cite{K1}. Specifically, we prove the following strengthening of a result from \cite{C}: For an exponentially closed ordinal $\alpha$, we have $L_{\alpha}\models$ZF$^{-}$ if and only if COMP$^{\text{ITRM}}_{\alpha}=L_{\alpha+1}\cap\mathfrak{P}(\alpha)$, i.e. if and only if the set of $\alpha$-ITRM-computable subsets of $\alpha$ coincides with the set of subsets of $\alpha$ in $L_{\alpha+1}$. Moreover, we show that, if $\alpha$ is exponentially closed and $L_{\alpha}\not\models$ZF$^{-}$, then COMP$^{\text{ITRM}}_{\alpha}=L_{\beta(\alpha)}\cap\mathfrak{P}(\alpha)$, where $\beta(\alpha)$ is the supremum of the $\alpha$-ITRM-clockable ordinals, which coincides with the supremum of the $\alpha$-ITRM-computable ordinals. We also determine the set of subsets of $\alpha$ computable by an $\alpha$-ITRM with time bounded below $\delta$ when $\delta>\alpha$ is an exponentially closed ordinal smaller than the supremum of the $\alpha$-ITRM-clockable ordinals. Moreover, we obtain some sufficient and necessary conditions on ordinals $\alpha$ for which the $\alpha$-wITRM-clockable ordinals are bounded by $\alpha$.