  The closure of deterministic context-free languages under logarithmic-space
many-one reductions ($\mathrm{L}$-m-reductions), known as LOGDCFL, has been
studied in depth from an aspect of parallel computability because it is nicely
situated between $\mathrm{L}$ and $\mathrm{AC}^{1}\cap\mathrm{SC}^2$. By
replacing a memory device from pushdown stacks with access-controlled storage
tapes, we introduce a computational model of one-way deterministic depth-$k$
storage automata ($k$-sda's) whose tape cells are freely modified during the
first $k$ accesses and then become blank forever. These $k$-sda's naturally
induce the language family $k\mathrm{SDA}$. Similarly to $\mathrm{LOGDCFL}$, we
study the closure $\mathrm{LOG}k\mathrm{SDA}$ of all languages in
$k\mathrm{SDA}$ under $\mathrm{L}$-m-reductions. We demonstrate that
$\mathrm{DCFL}\subseteq k\mathrm{SDA}\subseteq \mathrm{SC}^k$ by significantly
extending Cook's early result (1979) of $\mathrm{DCFL}\subseteq \mathrm{SC}^2$.
The entire hierarch of $\mathrm{LOG}k\mathrm{SDA}$ for all $k\geq1$ therefore
lies between $\mathrm{LOGDCFL}$ and $\mathrm{SC}$. As an immediate consequence,
we obtain the same simulation bounds for Hibbard's limited automata. We further
characterize $\mathrm{LOG}k\mathrm{SDA}$ in terms of a new machine model,
called logarithmic-space deterministic auxiliary depth-$k$ storage automata
that run in polynomial time. These machines are as powerful as a
polynomial-time two-way multi-head deterministic depth-$k$ storage automata. We
also provide a ``generic'' $\mathrm{LOG}k\mathrm{SDA}$-complete language under
$\mathrm{L}$-m-reductions by constructing a two-way universal simulator working
for all $k$-sda's.
