Abstract
We propose a logical characterization of problems solvable in deterministic polylogarithmic time (\(\mathrm {PolylogTime}\)). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. In the course of proving that our logic indeed captures \(\mathrm {PolylogTime}\) on finite ordered structures, we introduce a variant of random-access Turing machines that can access the relations and functions of the structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in \(\mathrm {PolylogTime}\).
The research reported in this paper results from the project Higher-Order Logics and Structures supported by the Austrian Science Fund (FWF: [I2420-N31]) and the Research Foundation Flanders (FWO: [G0G6516N]). It was further supported by the Austrian Research Promotion Agency (FFG) through the COMET funding for the Software Competence Center Hagenberg.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The term random access refers to the manner how random-access memory (RAM) is read and written. In contrast to sequential memory, the time it takes to read or write using RAM is almost independent of the physical location of the data in the memory. We want to emphasise that there is nothing random in random access.
- 2.
This ensures that \(F^\mathbf{A}_{\varphi , \bar{\mathtt {x}}, X}\) is monotonous and thus that the least fixed point \(\mathrm {lfp}(F^\mathbf{A}_{\varphi , \bar{\mathtt {x}}, X})\) is guaranteed to exists.
References
Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Boston (1995)
Ebbinghaus, H.D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1999)
Fagin, R.: Contributions to model theory of finite structures. Ph.D. thesis, U. C. Berkeley (1973)
Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. In: Karp, R. (ed.) Complexity of Computation. SIAM-AMS Proceedings, vol. 7, pp. 43–73 (1974)
Ferrarotti, F., González, S., Schewe, K.D., Turull Torres, J.M.: The polylog-time hierarchy captured by restricted second-order logic. In: Post-Proceedings of the 20th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. IEEE Computer Society (2019, to appear)
Ferrarotti F., González S., Turull Torres J.M., Van den Bussche J., Virtema J.: Descriptive complexity of deterministic polylogarithmic time. CoRR abs/1903.03413 (2019)
Grädel, E., et al.: Finite Model Theory and Its Applications. Springer, Heidelberg (2007). https://doi.org/10.1007/3-540-68804-8
Grandjean, E., Olive, F.: Graph properties checkable in linear time in the number of vertices. J. Comput. Syst. Sci. 68, 546–597 (2004)
Grohe, M.: Descriptive Complexity, Canonisation, and Definable Graph Structure Theory. Lecture Notes in Logic. Cambridge University Press, Cambridge (2017)
Grohe, M., Pakusa, W.: Descriptive complexity of linear equation systems and applications to propositional proof complexity. In: Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, pp. 1–12. IEEE Computer Society (2017)
Gurevich, Y.: Toward logic tailored for computational complexity. In: Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W. (eds.) Computation and Proof Theory. LNM, vol. 1104, pp. 175–216. Springer, Heidelberg (1984). https://doi.org/10.1007/BFb0099486
Gurevich, Y., Shelah, S.: Fixed-point extensions of first-order logic. Ann. Pure Appl. Logic 32, 265–280 (1986)
Immerman, N.: Number of quantifiers is better than number of tape cells. J. Comput. Syst. Sci. 22(3), 384–406 (1981)
Immerman, N.: Relational queries computable in polynomial time. Inf. Control 68, 86–104 (1986)
Immerman, N.: Descriptive Complexity. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0539-5
Knuth, D.: The Art of Computer Programming. Sorting and Searching, vol. 3. Addison-Wesley, Boston (1998)
Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-662-07003-1
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)
Ramakrishnan, R., Gehrke, J.: Database Management Systems, 3rd edn. McGraw-Hill, Inc., New York (2003)
Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976)
Vardi, M.: The complexity of relational query languages. In: Proceedings 14th ACM Symposium on the Theory of Computing, pp. 137–146 (1982)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer-Verlag GmbH Germany, part of Springer Nature
About this paper
Cite this paper
Ferrarotti, F., González, S., Turull Torres, J.M., Van den Bussche, J., Virtema, J. (2019). Descriptive Complexity of Deterministic Polylogarithmic Time. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science(), vol 11541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59533-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-662-59533-6_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-59532-9
Online ISBN: 978-3-662-59533-6
eBook Packages: Computer ScienceComputer Science (R0)