Hausdorff Reductions and the Exponential Hierarchies

Abstract

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.