Abstract
The polylogarithmic time hierarchy structures sub-linear time complexity. In recent work it was shown that all classes \(\tilde{\varSigma }_{m}^{ plog }\) or \(\tilde{\varPi }_{m}^{ plog }\) (\(m \in \mathbb {N}\)) 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.
The research reported in this paper results from the project Higher-Order Logics and Structures supported by the Austrian Science Fund (FWF: [I2420-N31]). It has also been partly supported by the Austrian Ministry for Transport, Innovation and Technology, the Federal Ministry for Digital and Economic Affairs, and the Province of Upper Austria in the frame of the COMET center SCCH.
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Notes
- 1.
This relationship appears quite natural in view of the well known relationship \(\mathrm {NP} = \mathrm {NTIME}(n^{O(1)}) \subseteq \mathrm {DTIME}(2^{{n}^{O(1)}}) = \mathrm {EXPTIME}\).
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Ferrarotti, F., González, S., Schewe, KD., Turull-Torres, J.M. (2020). Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems. In: Herzig, A., Kontinen, J. (eds) Foundations of Information and Knowledge Systems. FoIKS 2020. Lecture Notes in Computer Science(), vol 12012. Springer, Cham. https://doi.org/10.1007/978-3-030-39951-1_6
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