Completeness in Polylogarithmic Time and Space

TL;DR

This paper investigates the concept of completeness within polylogarithmic time and space complexity classes, proposing new notions of completeness and demonstrating the existence of complete problems under these frameworks.

Contribution

It introduces an alternative notion of completeness inspired by circuit uniformity and proves the existence of complete problems for PolylogSpace under this new definition.

Findings

Existence of proper hierarchies within polylogarithmic classes.

Development of a new notion of completeness based on uniformity.

Proof of a complete problem for PolylogSpace under the new notion.

Abstract

Complexity theory can be viewed as the study of the relationship between computation and applications, understood the former as complexity classes and the latter as problems. Completeness results are clearly central to that view. Many natural algorithms resulting from current applications have polylogarithmic time (PolylogTime) or space complexity (PolylogSpace). The classical Karp notion of complete problem however does not plays well with these complexity classes. It is well known that PolylogSpace does not have complete problems under logarithmic space many-one reductions. In this paper we show similar results for deterministic and non-deterministic PolylogTime as well as for every other level of the polylogarithmic time hierarchy. We achieve that by following a different strategy based on proving the existence of proper hierarchies of problems inside each class. We then develop an alternative notion of completeness inspired by the concept of uniformity from circuit complexity and prove the existence of a (uniformly) complete problem for PolylogSpace under this new notion. As a consequence of this result we get that complete problems can still play an important role in the study of the interrelationship between polylogarithmic and other classical complexity classes.