Function Spaces for Second-Order Polynomial Time

  • Akitoshi Kawamura
  • Arno Pauly
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8493)


In the context of second-order polynomial-time computability, we prove that there is no general function space construction. We proceed to identify restrictions on the domain or the codomain that do provide a function space with polynomial-time function evaluation containing all polynomial-time computable functions of that type.

As side results we show that a polynomial-time counterpart to admissibility of a representation is not a suitable criterion for natural representations, and that the Weihrauch degrees embed into the polynomial-time Weihrauch degrees.


cartesian closed computational complexity higher order computable analysis admissible representation Weihrauch reducibility 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauer, A.: Realizability as the connection between computable and constructive mathematics. Tutorial at CCA 2004, notes (2004)Google Scholar
  2. 2.
    Bauer, A.: A relationship between equilogical spaces and type two effectivity. Mathematical Logic Quarterly 48(1), 1–15 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., Pitassi, T.: The relative complexity of NP search problems. Journal of Computer and System Science 57, 3–19 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Annals of Pure and Applied Logic 163(8), 968–1008 (2012)CrossRefGoogle Scholar
  5. 5.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic 1, 73–117 (2011), arXiv:0905.4685Google Scholar
  6. 6.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. Journal of Symbolic Logic 76, 143–176 (2011), arXiv:0905.4679Google Scholar
  7. 7.
    Escardó, M.: Synthetic topology of datatypes and classical spaces. Electronic Notes in Theoretical Computer Science 87 (2004)Google Scholar
  8. 8.
    Férée, H., Gomaa, W., Hoyrup, M.: Analytical properties of resource-bounded real functionals. Journal of Complexity (to appear)Google Scholar
  9. 9.
    Férée, H., Hoyrup, M.: Higher-order complexity in analysis. In: CCA 2013 (2013)Google Scholar
  10. 10.
    Hennie, F.C., Stearns, R.E.: Two-tape simulation of multitape turing machines. J. ACM 13(4), 533–546 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Higuchi, K., Pauly, A.: The degree-structure of Weihrauch-reducibility. Logical Methods in Computer Science 9(2) (2013)Google Scholar
  12. 12.
    Kapron, B.M., Cook, S.A.: A new characterization of type-2 feasibility. SIAM Journal on Computing 25(1), 117–132 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Computational Complexity 19(2), 305–332 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kawamura, A.: On function spaces and polynomial-time computability. Dagstuhl Seminar 11411 (2011)Google Scholar
  15. 15.
    Kawamura, A., Cook, S.: Complexity theory for operators in analysis. ACM Transactions on Computation Theory 4(2), Article 5 (2012)Google Scholar
  16. 16.
    Kawamura, A., Müller, N., Rösnick, C., Ziegler, M.: Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime. arXiv 1211.4974 (2012)Google Scholar
  17. 17.
    Kawamura, A., Ota, H., Rösnick, C., Ziegler, M.: Computational complexity of smooth differential equations. Logical Methods in Computer Science 10(1), Paper 6 (2014)Google Scholar
  18. 18.
    Kawamura, A., Pauly, A.: Function spaces for second-order polynomial time. arXiv 1401.2861 (2014)Google Scholar
  19. 19.
    Ko, K.I.: Polynomial-time computability in analysis. Birkhäuser (1991)Google Scholar
  20. 20.
    Kreitz, C., Weihrauch, K.: Theory of representations. Theoretical Computer Science 38, 35–53 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Pauly, A.: Many-one reductions between search problems. arXiv 1102.3151 (2011),
  22. 22.
    Pauly, A.: Multi-valued functions in computability theory. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) CiE 2012. LNCS, vol. 7318, pp. 571–580. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Pauly, A.: A new introduction to the theory of represented spaces (2012),
  24. 24.
    Pauly, A., de Brecht, M.: Towards synthetic descriptive set theory: An instantiation with represented spaces. arXiv 1307.1850Google Scholar
  25. 25.
    Pauly, A., Ziegler, M.: Relative computability and uniform continuity of relations. Journal of Logic and Analysis 5 (2013)Google Scholar
  26. 26.
    Rösnick, C.: Closed sets and operators thereon. In: CCA 2013 (2013)Google Scholar
  27. 27.
    Schröder, M.: Admissible Representations for Continuous Computations. Ph.D. thesis, FernUniversität Hagen (2002)Google Scholar
  28. 28.
    Schröder, M.: Extended admissibility. Theoretical Computer Science 284(2), 519–538 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the LMS 2(42), 230–265 (1936)MathSciNetGoogle Scholar
  30. 30.
    Turing, A.: On computable numbers, with an application to the Entscheidungsproblem: Corrections. Proceedings of the LMS 2(43), 544–546 (1937)MathSciNetGoogle Scholar
  31. 31.
    Weihrauch, K.: Type 2 recursion theory. Theoretical Computer Science 38, 17–33 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Weihrauch, K.: Computable Analysis. Springer (2000)Google Scholar
  33. 33.
    Weihrauch, K.: Computational complexity on computable metric spaces. Mathematical Logic Quarterly 49(1), 3–21 (2003)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Akitoshi Kawamura
    • 1
  • Arno Pauly
    • 2
  1. 1.Department of Computer ScienceUniversity of TokyoJapan
  2. 2.Clare College, University of CambridgeUnited Kingdom

Personalised recommendations