Skip to main content
SpringerLink
Log in

Computability and Analysis, a Historical Approach

  • Conference paper
  • First Online:
Pursuit of the Universal (CiE 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9709))

Included in the following conference series:

Abstract

The history of computability theory and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, Émil Borel discussed his ideas on computable real number functions in his introduction to measure theory. On the other hand, Alan Turing had computable real numbers in mind when he introduced his now famous machine model. Here we want to focus on a particular aspect of computability and analysis, namely on computability properties of theorems from analysis. This is a topic that emerged already in early work of Turing, Specker and other pioneers of computable analysis and eventually leads us to the very recent project of classifying the computational content of theorems in the Weihrauch lattice.

V. Brattka—Supported by the National Research Foundation of South Africa. This article uses some historical insights that were established in [2].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See [19] for a definition of computably compact.

References

  1. Aberth, O.: The failure in computable analysis of a classical existence theorem for differential equations. Proc. Am. Math. Soc. 30, 151–156 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avigad, J., Brattka, V.: Computability and analysis: the legacy of Alan Turing. In: Downey, R. (ed.) Turing’s Legacy: Developments from Turing’s Ideas in Logic, LNL, vol. 42, pp. 1–47. Cambridge University Press, Cambridge, UK (2014)

    Chapter  Google Scholar 

  3. Baigger, G.: Die Nichtkonstruktivität des Brouwerschen Fixpunktsatzes. Arch. Math. Logic 25, 183–188 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beeson, M.J.: Foundations of Constructive Mathematics. A Series of Modern Surveys in Mathematics. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  5. Borel, É.: Le calcul des intégral définies. Journal de Mathematiques pures et appliquées 8(2), 159–210 (1912)

    MathSciNet  MATH  Google Scholar 

  6. Borel, É.: La théorie de la mesure et al théorie de l’integration. In: Leçons sur la théorie des fonctions, pp. 214–256. Gauthier-Villars, Paris, 4 edn. (1950)

    Google Scholar 

  7. Brattka, V.: Computability of Banach space principles. Informatik Berichte 286, FernUniversität Hagen, Fachbereich Informatik, Hagen, June 2001

    Google Scholar 

  8. Brattka, V.: Effective Borel measurability and reducibility of functions. Math. Logic Q. 51(1), 19–44 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brattka, V.: Computable versions of the uniform boundedness theorem. In: Chatzidakis, Z., et al. (eds.) Logic Colloquium 2002. LNL, vol. 27, pp. 130–151. ASL, Urbana (2006)

    Google Scholar 

  10. Brattka, V.: Borel complexity and computability of the Hahn-Banach Theorem. Arch. Math. Logic 46(7–8), 547–564 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brattka, V.: A computable version of Banach’s inverse mapping theorem. Ann. Pure Appl. Logic 157, 85–96 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Ann. Pure Appl. Logic 163, 986–1008 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symbolic Logic 17(1), 73–117 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symbolic Logic 76(1), 143–176 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brattka, V., Gherardi, G., Hölzl, R.: Probabilistic computability and choice. Inf. Comput. 242, 249–286 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brattka, V., Gherardi, G., Marcone, A.: The Bolzano-Weierstrass theorem is the jump of weak Kőnig’s lemma. Ann. Pure Appl. Logic 163, 623–655 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., et al. (eds.) New Computational Paradigms: Changing Conceptions of What is Computable, pp. 425–491. Springer, New York (2008)

    Chapter  Google Scholar 

  18. Brattka, V., Le Roux, S., Pauly, A.: Connected choice and the Brouwer fixed point theorem. arXiv 1206.4809. http://arxiv.org/abs/1206.4809

  19. Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theor. Comput. Sci. 305, 43–76 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Brattka, V., Yoshikawa, A.: Towards computability of elliptic boundary value problems in variational formulation. J. Complex. 22(6), 858–880 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gherardi, G., Marcone, A.: How incomputable is the separable Hahn-Banach theorem? Notre Dame Journal of Formal Logic 50(4), 393–425 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grzegorczyk, A.: Computable functionals. Fundamenta Mathematicae 42, 168–202 (1955)

    MathSciNet  MATH  Google Scholar 

  23. Hertling, P.: An effective Riemann mapping theorem. Theor. Comput. Sci. 219, 225–265 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hoyrup, M., Rojas, C., Weihrauch, K.: Computability of the Radon-Nikodym derivative. Computability 1(1), 3–13 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Jockusch, C.G., Soare, R.I.: \(\Pi ^{0}_{1}\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)

    MathSciNet  MATH  Google Scholar 

  26. Kleene, S.C.: Recursive functions and intuitionistic mathematics. In: Proceedings of the International Congress of Mathematicians, Cambridge, 1950, vol. 1, pp. 679–685. AMS, Providence (1952)

    Google Scholar 

  27. Ko, K.I.: Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston (1991)

    Book  MATH  Google Scholar 

  28. Kreisel, G.: On the interpretation of non-finitist proofs, II: interpretation of number theory, applications. J. Symbolic Logic 17, 43–58 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lacombe, D.: Remarques sur les opérateurs récursifs et sur les fonctions récursives d’une variable réelle. C. R. Acad. Paris 241, 1250–1252, théorie des fonctions, November 1955

    Google Scholar 

  30. Lacombe, D.: Les ensembles récursivement ouverts ou fermés, et leurs applications à l’Analyse récursive. C. R. Acad. Paris 245, 1040–1043, logique (1957)

    Google Scholar 

  31. Le Roux, S., Ziegler, M.: Singular coverings and non-uniform notions of closed set computability. Math. Logic Q. 54(5), 545–560 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Metakides, G., Nerode, A.: The introduction of non-recursive methods into mathematics. In: Troelstra, A., van Dalen, D. (eds.) The L.E.J. Brouwer Centenary Symposium, pp. 319–335. North-Holland, Amsterdam (1982)

    Google Scholar 

  33. Metakides, G., Nerode, A., Shore, R.A.: Recursive limits on the Hahn-Banach theorem. In: Rosenblatt, M. (ed.) Errett Bishop: Reflections on Him and His Research. AMS, Providence (1985). Contemp. Math. vol. 39, pp. 85–91

    Google Scholar 

  34. Orevkov, V.P.: A constructive mapping of the square onto itself displacing every constructive point (Russian). Doklady Akademii Nauk 152, 55–58 (1963). Translated in: Soviet Math. - Dokl. vol. 4, pp. 1253–1256 (1963)

    MathSciNet  MATH  Google Scholar 

  35. Pauly, A.: How incomputable is finding Nash equilibria? J. Univers. Comput. Sci. 16(18), 2686–2710 (2010)

    MathSciNet  MATH  Google Scholar 

  36. Pauly, A.: On the (semi)lattices induced by continuous reducibilities. Math. Logic Q. 56(5), 488–502 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pour-El, M.B., Richards, J.I.: A computable ordinary differential equation which possesses no computable solution. Ann. Math. Logic 17, 61–90 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  39. Rice, H.G.: Recursive real numbers. Proc. Am. Math. Soc. 5, 784–791 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  40. Simpson, S.G.: Subsystems of Second Order Arithmetic, 2nd edn. Cambridge University Press, Poughkeepsie (2009)

    Book  MATH  Google Scholar 

  41. Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symbolic Logic 14(3), 145–158 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  42. Specker, E.: Der Satz vom Maximum in der rekursiven Analysis. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 254–265. North-Holland, Amsterdam (1959)

    Google Scholar 

  43. Turing, A.M.: On computable numbers, with an application to the “Entscheidungsproblem”. Proc. London Math. Soc. 42(2), 230–265 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  44. Turing, A.M.: On computable numbers, with an application to the “Entscheidungsproblem”. Correction Proc. London Math. Soc. 43(2), 544–546 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  45. Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  46. Zaslavskiĭ, I.D.: Disproof of some theorems of classical analysis in constructive analysis (Russian). Usp. Mat. Nauk 10(4), 209–210 (1955)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computer Science, Universität der Bundeswehr München, München, Germany

    Vasco Brattka

  2. Deptartment of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa

    Vasco Brattka

Authors
  1. Vasco Brattka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vasco Brattka .

Editor information

Editors and Affiliations

  1. Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. Université Paris-Diderot - Paris 7, Paris, France

    Laurent Bienvenu

  3. University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Brattka, V. (2016). Computability and Analysis, a Historical Approach. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-40189-8_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40188-1

  • Online ISBN: 978-3-319-40189-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation