  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
$\alpha$.
