Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems

TL;DR

This paper explores the structure of the polylogarithmic time hierarchy, demonstrating strict hierarchies within classes and proving the absence of complete problems, thereby advancing the understanding of sub-linear time complexity.

Contribution

It establishes the strictness of hierarchies within polylogarithmic time classes and proves there are no complete problems for these classes, deepening descriptive complexity theory.

Findings

Hierarchies within polylogarithmic time classes are strict

No complete problems exist for these classes

Polylogarithmic time hierarchy itself is strict

Abstract

The polylogarithmic time hierarchy structures sub-linear time complexity. In recent work it was shown that all classes $\tilde{\Sigma}_{m}^{\mathit{plog}}$ or $\tilde{\Pi}_{m}^{\mathit{plog}}$ ($m \in \mathbb{N}$) in this hierarchy can be captured by semantically restricted fragments of second-order logic. In this paper the descriptive complexity theory of polylogarithmic time is taken further showing that there are strict hierarchies inside each of the classes of the hierarchy. A straightforward consequence of this result is that there are no complete problems for these complexity classes, not even under polynomial time reductions. As another consequence we show that the polylogarithmic time hierarchy itself is strict.