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.

中文翻译：

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多对数时间和空间的完备性

复杂性理论可以看作是对计算和应用之间关系的研究，将前者理解为复杂度类，后者理解为问题。完整性结果显然是该观点的核心。当前应用程序产生的许多自然算法具有多对数时间 (PolylogTime) 或空间复杂度 (PolylogSpace)。然而，完整问题的经典 Karp 概念不适用于这些复杂性类别。众所周知，PolylogSpace 在对数空间多一约简下不存在完全问题。在本文中，我们展示了确定性和非确定性 PolylogTime 以及多对数时间层次结构的每个其他级别的类似结果。我们通过遵循基于证明每个类中问题的适当层次结构存在的不同策略来实现这一点。然后，我们从电路复杂性的一致性概念中得到启发，开发了另一种完备性概念，并证明在这个新概念下 PolylogSpace 存在一个（一致）完备问题。由于这个结果，我们得到完整问题仍然可以在研究多对数和其他经典复杂性类之间的相互关系中发挥重要作用。