  This paper extends our paper \cite{C2} for the conference ``Computability in
Europe'' 2022.
  After Infinite Time Turing Machines (ITTM) were introduced in Hamkins and
Lewis \cite{HL}, a number of machine models of computability have been
generalized to the transfinite, along with various variants thereof. While for
some of these models the computational strength has been successfully
determined, there are still several white spots on the map of transfinite
computability. In this paper, we contribute to the understanding of the
computational strength of transfinite machine models by (i) proving lower
bounds on the computational strength of $\alpha$-Infinite Time Register
Machines ($\alpha$-ITRMs) for certain values of $\alpha$, refuting a conjecture
about their strength made in \cite{alpha itrms}, (ii) showing that the
computational strength of cardinal-recognizing ITRMs is equal to that of ITRMs
and (iii) showing that non-solvability of the bounded halting problem,
existence of a universal machine and an increase of computational power by
allowing machines to recognize cardinals are equivalent for $\alpha$-ITRMs for
all relevant values of $\alpha$ .
