Two double-exponential gaps for automata with a limited pushdown

We shall consider nondeterministic and deterministic automata equipped with a limited pushdown (constant height npdas and dpdas) as well as their two-way versions (constant height 2npdas and 2dpdas). We show two double-exponential gaps for these devices, namely, (i) for complementing constant height one-way npdas and (ii) for converting constant height 2npdas or 2dpdas into one-way devices.