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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
This relationship appears quite natural in view of the well known relationship \(\mathrm {NP} = \mathrm {NTIME}(n^{O(1)}) \subseteq \mathrm {DTIME}(2^{{n}^{O(1)}}) = \mathrm {EXPTIME}\).
References
Babai, L.: Graph isomorphism in quasipolynomial time. In: Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing (STOC 2016), pp. 684–697 (2016)
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic. Springer, Heidelberg (1995)
Ferrarotti, F., González, S., Schewe, K.-D., Turull Torres, J.M.: The polylog-time hierarchy captured by restricted second-order logic. In: 20th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2018, Timisoara Romania, 20–23 September 2018, pp. 133–140. IEEE (2018)
Ferrarotti, F., González, S., Schewe, K.-D., Turull Torres, J.M.: The polylog-time hierarchy captured by restricted second-order logic. CoRR, abs/1806.07127 (2018)
Ferrarotti, F., González, S., Schewe, K.-D., Turull Torres, J.M.: A restricted second-order logic for non-deterministic poly-logarithmic time. Logic J. IGPL (2019, to appear)
Ferrarotti, F., González, S., Turull Torres, J.M., Van den Bussche, J., Virtema, J.: Descriptive complexity of deterministic polylogarithmic time. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds.) WoLLIC 2019. LNCS, vol. 11541, pp. 208–222. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-662-59533-6_13
Ferrarotti, F., González, S., Turull Torres, J.M., Van den Bussche, J., Virtema, J.: Descriptive complexity of deterministic polylogarithmic time and space. Submitted for publication (2019)
Immerman, N.: Descriptive complexity. Graduate Texts in Computer Science. Springer, New York (1999)
Mix Barrington, D.A.: Quasipolynomial size circuit classes. In: Proceedings of the Seventh Annual Structure in Complexity Theory Conference, Boston, Massachusetts, USA, 22–25 June 1992, pp. 86–93. IEEE Computer Society (1992)
Mix Barrington, D.A., Immerman, N., Straubing, H.: On uniformity within NC\(^1\). J. Comput. Syst. Sci. 41(3), 274–306 (1990)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-39951-1_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-39950-4
Online ISBN: 978-3-030-39951-1
eBook Packages: Computer ScienceComputer Science (R0)