多对数时空中的完备性
Completeness in Polylogarithmic Time and Space
摘要：
复杂性理论可以被看作是对计算和应用之间关系的研究，将前者理解为复杂性类，后者理解为问题。完整性结果显然是这一观点的核心。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，经典的卡普完全问题概念并不能很好地处理这些复杂性类。众所周知，多对数空间在对数空间多1约简下不存在完全问题。在本文中，我们显示了类似的结果，确定性和非确定性的PolylogTime以及其他级别的polylogarithmic时间层次。我们通过遵循不同的策略来实现这一点，该策略基于证明每个类内部存在适当的问题层次。然后，我们开发了另一种概念的完整性的启发一致性的概念，从电路复杂性，并证明了一个（一致）完整的问题的存在下，这个新的概念。由于这一结果，我们得到的完整的问题仍然可以发挥重要作用，在研究之间的相互关系polylogarithmic和其他经典的复杂性类。
英文摘要：
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.
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