  The complexity of several logics, such as Presburger arithmetic, dependence
logics and ambient logics, can only be characterised in terms of alternating
Turing machines. Despite quite natural, the presence of alternation can
sometimes cause neat ideas to be obfuscated inside heavy technical machinery.
In these notes, we propose two problems on deterministic machines that can be
used to prove lower bounds with respect to the computational class
$k$AExp$_{\text{pol}}$, that is the class of all problems solvable by an
alternating Turing machine running in $k$ exponential time and performing a
polynomial amount of alternations, with respect to the input size. The first
problem, called $k$AExp$_{\text{pol}}$-prenex TM problem, is a problem about
deterministic Turing machines. The second problem, called the $k$-exp
alternating multi-tiling problem, is analogous to the first one, but on tiling
systems.
  Both problems are natural extensions of the TM alternation problem and the
alternating multi-tiling problem proved AExp$_{\text{pol}}$-complete by L.
Bozzelli, A. Molinari, A. Montanari and A. Peron in [GandALF, pp. 31-45, 2017].
The proofs presented in these notes follow the elegant exposition in A.
Molinari's PhD thesis to extend these results from the case $k = 1$ to the case
of arbitrary $k$.
