On the Computational Complexity of Positive Linear Functionals on \(\mathcal{C}[0;1]\)

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


The Lebesgue integration has been related to polynomial counting complexity in several ways, even when restricted to smooth functions. We prove analogue results for the integration operator associated with the Cantor measure as well as a more general second-order \({{\mathbf {\mathsf{{\#P}}}}} \)-hardness criterion for such operators. We also give a simple criterion for relative polynomial time complexity and obtain a better understanding of the complexity of integration operators using the Lebesgue decomposition theorem.


  1. [BrGh11]
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symb. Log. 76(1), 143–176 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. [Coll14]
    Collins, P.: Computable Stochastic Processes (2014). arXiv:1409.4667
  3. [Cook91]
    Cook, S.A.: Computability and complexity of higher type functions. In: Moschovakis, Y.N. (ed.) Logic from Computer Science. Mathematical Sciences Research Institute Publications, pp. 51–72. Springer, Heidelberg (1991)Google Scholar
  4. [FeHo13]
    Férée, H., Hoyrup, M.: Higher-order complexity in analysis. In: Proceedings 10th International Conference on Computability and Complexity in Analysis (CCA 2013)Google Scholar
  5. [FGH13]
    Férée, H., Gomaa, W., Hoyrup, M.: Analytical properties of resource-bounded real functionals. J. Complex. 30(5), 647–671 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. [Frie84]
    Friedman, H.: The computational complexity of maximization and integration. Adv. Math. 53, 80–98 (1984)MathSciNetCrossRefMATHGoogle Scholar
  7. [HeOg02]
    Hemaspaandra, L.A., Ogihara, M.: The Complexity Theory Companion. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
  8. [HiPa13]
    Higuchi, K., Pauly, A.: The degree structure of Weihrauch-reducibility. Log. Methods Comput. Sci. 9(2), 1–17 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. [HRW12]
    Hoyrup, M., Rojas, C., Weihrauch, K.: Computability of the Radon-Nikodym derivative. Computability 1, 1–11 (2012)MathSciNetMATHGoogle Scholar
  10. [IBR01]
    Irwin, R., Kapron, B., Royer, J.: On characterizations of the basic feasible functionals part I. J. Funct. Program. 11, 117–153 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. [KaCo96]
    Kapron, B.M., Cook, S.A.: A new characterization of type-2 feasibility. SIAM J. Comput. 25(1), 117–132 (1996)MathSciNetCrossRefMATHGoogle Scholar
  12. [KaCo10]
    Kawamura, A., Cook, S.A.: "Complexity theory for operators in analysis. In: Proceedings of 42nd Annual ACM Symposium on Theory of Computing (STOC 2010), pp. 495–502 (2012). (full version in ACM Transactions in Computation Theory, vol. 4:2 , article 5.)Google Scholar
  13. [KaPa14]
    Kawamura, A., Pauly, A.: Function spaces for second-order polynomial time. In: Beckmann, A., Csuhaj-Varjú, E., Meer, K. (eds.) CiE 2014. LNCS, vol. 8493, pp. 245–254. Springer, Heidelberg (2014)Google Scholar
  14. [Kawa11]
    Kawamura, A.: Computational complexity in analysis and geometry, Dissertation, University of Toronto (2011)Google Scholar
  15. [Ko91]
    Ko, K.-I.: Computational Complexity of Real Functions. Birkhäuser, Boston (1991)CrossRefMATHGoogle Scholar
  16. [KSZ14]
    Kawamura, A., Steinberg, F., Ziegler, M.: Complexity of Laplace’s and Poisson’s Equation, abstract. Bull. Symb. Log. 20(2), 231 (2014). Full version to appear in Logical Methods in Computer ScienceGoogle Scholar
  17. [KSZ15]
    Kawamura, A., Steinberg, F., Ziegler, M.: Computational Complexity Theory for classes of integrable functions. In: JAIST Logic Workshop Series (2015)Google Scholar
  18. [MTY14]
    Mori, T., Tsujii, Y., Yasugi, M.: Computability of probability distributions and characteristic functions. Log. Methods Comput. Sci. 9, 3 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. [Schr07]
    Schröder, M.: Admissible representations of probability measures. Electron. Notes Theoret. Comput. Sci. 167, 61–78 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. [Weih00]
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsTU DarmstadtDarmstadtGermany
  2. 2.Université de Lorraine, LORIA, UMR 7503Vandœuvre-lès-NancyFrance
  3. 3.InriaVillers-lès-NancyFrance
  4. 4.CNRS, LORIA, UMR 7503Vandœuvre-lès-NancyFrance
  5. 5.KAIST, School of ComputingDaejeonRepublic of Korea

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