Second-Order Linear-Time Computability with Applications to Computable Analysis

  • Akitoshi Kawamura
  • Florian Steinberg
  • Holger ThiesEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)


In this work we put forward a complexity class of type-two linear-time. For such a definition to be meaningful, a detailed protocol for the cost of interactions with functional inputs has to be fixed. This includes some design decisions the defined class is sensible to and we carefully discuss our choices and their implications. We further discuss some properties and examples of operators that are and are not computable in linear-time and nearly linear-time and some applications to computable analysis.


Computational complexity Linear time Computable analysis 



This work was supported by JSPS KAKENHI Grant Numbers JP18H03203 and JP18J10407, by the Japan Society for the Promotion of Science (JSPS), Core-to-Core Program (A. Advanced Research Networks), by the ANR project FastRelax (ANR-14-CE25-0018-01) of the French National Agency for Research and by EU-MSCA-RISE project 731143 “Computing with Infinite Data” (CID).


  1. 1.
    Brausse, F., Steinberg, F.: A minimal representation for continuous functions. arXiv preprint arXiv:1703.10044 (2017)
  2. 2.
    Buss, J.F.: Relativized alternation and space-bounded computation. J. Comput. Syst. Sci. 36(3), 351–378 (1988)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Case, J.: Resource restricted computability theoretic learning: illustrative topics and problems. Theory Comput. Syst. 45(4), 773–786 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Case, J., Kötzing, T., Paddock, T.: Feasible iteration of feasible learning functionals. In: Hutter, M., Servedio, R.A., Takimoto, E. (eds.) ALT 2007. LNCS (LNAI), vol. 4754, pp. 34–48. Springer, Heidelberg (2007). Scholar
  5. 5.
    Férée, H., Hainry, E., Hoyrup, M., Péchoux, R.: Interpretation of stream programs: characterizing type 2 polynomial time complexity. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 291–303. Springer, Heidelberg (2010). Scholar
  6. 6.
    Fournet, C., Kohlweiss, M., Strub, P.-Y.: Modular code-based cryptographic verification. In: Proceedings of the 18th ACM Conference on Computer and Communications Security, pp. 341–350. ACM (2011)Google Scholar
  7. 7.
    Grzegorczyk, A.: On the definitions of computable real continuous functions. Fund. Math. 44, 61–71 (1957)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gurevich, Y., Shelah, S.: Nearly linear time. In: Meyer, A.R., Taitslin, M.A. (eds.) Logic at Botik 1989. LNCS, vol. 363, pp. 108–118. Springer, Heidelberg (1989). Scholar
  9. 9.
    Irwin, R.J., Royer, J.S., Kapron, B.M.: On characterizations of the basic feasible functionals, part i. J. Funct. Program. 11(1), 117–153 (2001)CrossRefGoogle Scholar
  10. 10.
    Kapron, B.M., Cook, S.A.: A new characterization of type-2 feasibility. SIAM J. Comput. 25(1), 117–132 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kawamura, A., Cook, S.: Complexity theory for operators in analysis. ACM Trans. Comput. Theory 4(2), Article 5 (2012)Google Scholar
  12. 12.
    Kawamura, A., Ota, H.: Small complexity classes for computable analysis. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 432–444. Springer, Heidelberg (2014). Scholar
  13. 13.
    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, Cham (2014). Scholar
  14. 14.
    Kawamura, A., Steinberg, F., Thies, H.: Parameterized complexity for uniform operators on multidimensional analytic functions and ODE solving. In: Moss, L.S., de Queiroz, R., Martinez, M. (eds.) WoLLIC 2018. LNCS, vol. 10944, pp. 223–236. Springer, Heidelberg (2018). Scholar
  15. 15.
    Lacombe, D.: Sur les possibilités d’extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. In: Le raisonnement en mathématiques et en sciences expérimentales, Colloques Internationaux du Centre National de la Recherche Scientifique, LXX. Editions du Centre National de la Recherche Scientifique, Paris, pp. 67–75 (1958)Google Scholar
  16. 16.
    Mehlhorn, K.: Polynomial and abstract subrecursive classes. In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pp. 96–109. ACM (1974)Google Scholar
  17. 17.
    Pauly, A., Steinberg, F.: Comparing representations for function spaces in computable analysis. Theory Comput. Syst. 62(3), 557–582 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Regan, K.W.: Machine models and linear time complexity. ACM SIGACT News 24(3), 5–15 (1993)CrossRefGoogle Scholar
  19. 19.
    Schröder, M.: Extended admissibility. Theor. Comput. Sci. 284(2), 519–538 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Schröder, M.: Admissible representations for continuous computations. Fernuniv., Fachbereich Informatik (2003)Google Scholar
  21. 21.
    Steinberg, F.: Computational complexity theory for advanced function spaces in analysis. Ph.D. thesis, Technische Universität (2017)Google Scholar
  22. 22.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. 2(1), 230–265 (1936)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Weihrauch, K.: Computable Analysis. Springer, Berlin/Heidelberg (2000). Scholar
  24. 24.
    Ziegler, M.: Hyper-degrees of 2nd-order polynomial-time reductions. Measuring the Complexity of Computational Content: Weihrauch Reducibility and Reverse Analysis (Dagstuhl Seminar 15392)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Akitoshi Kawamura
    • 1
  • Florian Steinberg
    • 2
  • Holger Thies
    • 1
    Email author
  1. 1.Kyushu UniversityFukuokaJapan
  2. 2.Inria SaclayPalaiseauFrance

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