Taming Koepke's Zoo II: Register Machines

Abstract

We study the computational strength of resetting $α$-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 $α$, we have $L_α\models$ZF$^{-}$ if and only if COMP$^{\text{ITRM}}_α=L_{α+1}\cap\mathfrak{P}(α)$, i.e. if and only if the set of $α$-ITRM-computable subsets of $α$ coincides with the set of subsets of $α$ in $L_{α+1}$. Moreover, we show that, if $α$ is exponentially closed and $L_α\not\models$ZF$^{-}$, then COMP$^{\text{ITRM}}_α=L_{β(α)}\cap\mathfrak{P}(α)$, where $β(α)$ is the supremum of the $α$-ITRM-clockable ordinals, which coincides with the supremum of the $α$-ITRM-computable ordinals. We also determine the set of subsets of $α$ computable by an $α$-ITRM with time bounded below $δ$ when $δ>α$ is an exponentially closed ordinal smaller than the supremum of the $α$-ITRM-clockable ordinals. Moreover, we obtain some sufficient and necessary conditions on ordinals $α$ for which the $α$-wITRM-clockable ordinals are bounded by $α$.