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 $k$.

翻译：一些复杂的逻辑,如Presburger算术、依赖逻辑和环境逻辑等,只能以交替图灵机器的形式来描述。尽管相当自然,交替存在有时会使一些精美的想法在重型技术机械中被混淆。在这些注释中,我们提出了在确定性机器上的两个问题,这些机器可以用来证明计算级的较低界限$k$AExp${text{pol}$,这是所有问题的类别,因为交替的图灵机器以美元指数时间运行,并且对输入大小进行多盘化。第一个问题叫做$k$AExplat${text{pol_$-prenex TM问题,这是确定性图灵机的问题。第二个问题,称为$k美元交替的多盘化问题,类似于第一个问题,但关于平整流系统的问题。这两个问题都是TM变换的自然延伸结果,而交替的多盘调问题则是AExplain$_talari$A_pol_Q_G_Q_BAR_BAR_BAR_BAR_BAR_BAR_BAR_BAR_A_BAR_BAR_BAR_BAR_BAR_BAR_BAR_BAR_BAR_31.