We introduce Hausdorff (complexity) classes, which provide canonical characterizations of the intermediate levels of the iterated exponential hierarchies, including the Polynomial Hierarchy, the (Weak) Exponential Hierarchy, and higher-order exponential hierarchies. As certificates characterize main hierarchy levels without oracles, Hausdorff classes give an oracle-free characterization of intermediate hierarchy levels. The Hausdorff perspective provides a structural explanation for many known equivalences between oracle classes. In particular, seemingly different oracle classes corresponding to the same intermediate level are shown to arise from just three different, yet equivalent, oracle-aided approaches to deciding languages in a single Hausdorff class, thus replacing multiple oracle-based views with a unique characterization. It also explains the collapse of the Strong Exponential Hierarchy, showing that $\mathrm{P}^{\mathrm{NExp}} = \mathrm{NP}^{\mathrm{NExp}}$ arises because both classes coincide with the same Hausdorff class, thereby resolving a question of Hemachandra. Finally, we define canonical complete problems yielding matching lower bounds for $\mathrm{P}^{\mathrm{NExp[Log]}}$ problems whose hardness was left open due to the lack of known $\mathrm{P}^{\mathrm{NExp[Log]}}$-complete problems.

翻译：我们引入豪斯多夫（复杂性）类，为迭代指数层级（包括多项式层级、（弱）指数层级及高阶指数层级）的中间层级提供了规范刻画。与利用证书无预言机刻画主要层级不同，豪斯多夫类提供了中间层级无预言机的表征。豪斯多夫视角为预言机类之间的许多已知等价性提供了结构性解释。特别地，对应同一中间层级看似不同的预言机类，被证明源自对单个豪斯多夫类中语言判定的三种不同但等价的预言机辅助方法，从而以唯一刻画取代了多重预言机视角。该视角还解释了强指数层级的坍塌，表明$\mathrm{P}^{\mathrm{NExp}} = \mathrm{NP}^{\mathrm{NExp}}$的产生是由于这两个类与同一豪斯多夫类重合，由此解决了Hemachandra提出的问题。最后，我们定义了规范完全问题，为$\mathrm{P}^{\mathrm{NExp[Log]}}$问题给出匹配下界——此前因缺乏已知的$\mathrm{P}^{\mathrm{NExp[Log]}}$-完全问题，其难度问题尚未解决。