Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems

Flavio Ferrarotti; Senén González; Klaus-Dieter Schewe; José María Turull-Torres

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arXiv · 1911.13104

Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems

Flavio Ferrarotti·Senén González·Klaus-Dieter Schewe·José María Turull-Torres

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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.

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Flavio Ferrarotti, Senén González, Klaus-Dieter Schewe, José María Turull-Torres. 2019-11-29. Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems. https://arxiv.org/abs/1911.13104

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