The Strong Exponential Hierarchy $SEH$ was shown to collapse to $P^{NExp}$ by Hemachandra by proving $P^{NExp} = NP^{NExp}$ via a census argument. Nonetheless, Hemachandra also asked for certificate-based and alternating Turing machine characterizations of the $SEH$ levels, in the hope that these might have revealed deeper structural reasons behind the collapse. These open questions have thus far remained unanswered. To close them, by building upon the notion of Hausdorff reductions, we investigate a natural normal form for the intermediate levels of the (generalized) exponential hierarchies, i.e., the single-, the double-Exponential Hierarchy, and so on. Although the two characterizations asked for derive from our Hausdorff characterization, it is nevertheless from the latter that a surprising structural reason behind the collapse of $SEH$ is uncovered as a consequence of a very general result: the intermediate levels of the exponential hierarchies are precisely characterized by specific "Hausdorff classes", which define these levels without resorting to oracle machines. By this, contrarily to oracle classes, which may have different shapes for a same class (e.g., $P^{NP}_{||} = P^{NP[Log]} = LogSpace^{NP}$), hierarchy intermediate levels are univocally identified by Hausdorff classes (under the hypothesis of no hierarchy collapse). In fact, we show that the rather simple reason behind many equivalences of oracle classes is that they just refer to different ways of deciding the languages of a same Hausdorff class, and this happens also for $P^{NExp}$ and $NP^{NExp}$. In addition, via Hausdorff classes, we define complete problems for various intermediate levels of the exponential hierarchies. Through these, we obtain matching lower-bounds for problems known to be in $P^{NExp[Log]}$, but whose hardness was left open due to the lack of known $P^{NExp[Log]}$-complete problems.

翻译：强指数层级$SEH$曾被Hemachandra通过人口普查论证证明其坍缩至$P^{NExp}$，即$P^{NExp} = NP^{NExp}$。尽管如此，Hemachandra仍提出了对$SEH$层级基于证书和交替图灵机刻画的要求，期望这些刻画可能揭示该坍缩背后更深层的结构原因。这些开放问题迄今未获解答。为填补这一空白，基于Hausdorff归约的概念，我们研究了（广义）指数层级中间层级的自然范式，即单指数层级、双指数层级等。尽管所要求的两种刻画源于我们的Hausdorff刻画，但正是从后者中，揭示出$SEH$坍缩背后一个令人惊讶的结构原因——这源于一个非常一般的结果：指数层级的中间层级可由特定的“Hausdorff类”精确刻画，这些类无需借助预言机即可定义这些层级。由此，与预言机类（例如$P^{NP}_{||} = P^{NP[Log]} = LogSpace^{NP}$）可能对同一类呈现不同形态不同，层级中间层级由Hausdorff类唯一标识（在无层级坍缩的假设下）。事实上，我们表明许多预言机类等价性的相当简单的原因在于，它们仅指代判定同一Hausdorff类语言的不同方式，并且这也适用于$P^{NExp}$与$NP^{NExp}$。此外，通过Hausdorff类，我们为指数层级的各类中间层级定义了完全问题。借此，我们获得了已知属于$P^{NExp[Log]}$的问题的匹配下界，而由于此前缺乏已知的$P^{NExp[Log]}$完全问题，这些问题的困难性一直未解决。