The polylogarithmic time hierarchy structures sub-linear time complexity. In recent work it was shown that all classes (formula presented) or (formula presented) 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.
CITATION STYLE
Ferrarotti, F., González, S., Schewe, K. D., & Turull-Torres, J. M. (2020). Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 12012 LNCS, pp. 90–105). Springer. https://doi.org/10.1007/978-3-030-39951-1_6
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