The Brouwer Fixed Point Theorem Revisited

  • Vasco Brattka
  • Stéphane Le Roux
  • Joseph S. Miller
  • Arno PaulyEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)


We revisit the investigation of the computational content of the Brouwer Fixed Point Theorem in [7], and answer the two open questions from that work. First, we show that the computational hardness is independent of the dimension, as long as it is greater than 1 (in [7] this was only established for dimension greater than 2). Second, we show that restricting the Brouwer Fixed Point Theorem to L-Lipschitz functions for any \(L > 1\) also does not change the computational strength, which together with prior results establishes a trichotomy for \(L > 1\), \(L = 1\) and \(L < 1\).


  1. 1.
    Baigger, G.: Die Nichtkonstruktivität des Brouwerschen Fixpunktsatzes. Archiv für mathematische Logik und Grundlagenforschung 25, 183–188 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Ann. Pure Appl. Logic 163(8), 968–1008 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symbolic Logic 1, 73–117 (2011). arXiv:0905.4685 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symbolic Logic 76, 143–176 (2011). arXiv:0905.4679 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brattka, V., Gherardi, G., Marcone, A.: The Bolzano-Weierstrass theorem is the jump of weak König’s lemma. Ann. Pure Appl. Logic 163(6), 623–625 (2012). arXiv:1101.0792 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brattka, V., Le Roux, S., Pauly, A.: Connected choice and Brouwer’s fixed point theorem (2012).
  7. 7.
    Brattka, V., Le Roux, S., Pauly, A.: On the computational content of the brouwer fixed point theorem. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) CiE 2012. LNCS, vol. 7318, pp. 56–67. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Gherardi, G., Marcone, A.: How incomputable is the separable Hahn-Banach theorem? Notre Dame J. Formal Logic 50(4), 393–425 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hertling, P.: Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Ph.D. thesis, Fernuniversität, Gesamthochschule in Hagen, Oktober 1996Google Scholar
  10. 10.
    Higuchi, K., Pauly, A.: The degree-structure of Weihrauch-reducibility. Logical Meth. Comput. Sci. 9(2), 1–17 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hoyrup, M., Rojas, C., Weihrauch, K.: Computability of the Radon-Nikodym derivative. Computability 1(1), 3–13 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Le Roux, S., Pauly, A.: Finite choice, convex choice and finding roots. Logical Methods in Computer Science (2015).
  13. 13.
    Le Roux, S., Pauly, A.: Weihrauch degrees of finding equilibria in sequential games. In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 246–257. Springer, Heidelberg (2015). 25 CrossRefGoogle Scholar
  14. 14.
    Miller, J.S.: \(\Pi _1^0\) Classes in Computable Analysis and Topology. Ph.D. thesis. Cornell University (2002)Google Scholar
  15. 15.
    Neumann, E.: Computational problems in metric fixed point theory and their Weihrauch degrees. Logic. Methods Comput. Sci. 11(4) (2015)Google Scholar
  16. 16.
    Orevkov, V.: A constructive mapping of a square onto itself displacing every constructive point. Sov. Math. IV Trans. Doklady Akademie Nauk SSSR. 152(1), 55 (1963). Published by the American Mathematical SocietyMathSciNetzbMATHGoogle Scholar
  17. 17.
    Pauly, A.: How incomputable is finding Nash equilibria? J. Univers. Comput. Sci. 16(18), 2686–2710 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pauly, A.: On the (semi) lattices induced by continuous reducibilities. Math. Logic Q. 56(5), 488–502 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pauly, A.: On the topological aspects of the theory of represented spaces. Computability (2016).
  20. 20.
    Potgieter, P.H.: Computable counter-examples to the Brouwer fixed-point theorem (2008).
  21. 21.
    Shioji, N., Tanaka, K.: Fixed point theory in weak second-order arithmetic. Ann. Pure Appl. Logic 47, 167–188 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Simpson, S.: Subsystems of Second Order Arithmetic. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  23. 23.
    von Stein, T.: Vergleich nicht konstruktiv lösbarer Probleme in der Analysis. Diplomarbeit, Fachbereich Informatik. FernUniversität Hagen (1989)Google Scholar
  24. 24.
    Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Informatik Berichte 129, FernUniversität Hagen, Hagen, July 1992Google Scholar
  25. 25.
    Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen, September 1992Google Scholar
  26. 26.
    Weihrauch, K.: Computable Analysis. Springer-Verlag, New York (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Vasco Brattka
    • 1
    • 2
  • Stéphane Le Roux
    • 3
  • Joseph S. Miller
    • 4
  • Arno Pauly
    • 3
    Email author
  1. 1.Faculty of Computer ScienceUniversity of the Armed Forces MunichNeubibergGermany
  2. 2.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa
  3. 3.Département d’InformatiqueUniversité libre de BruxellesBrusselsBelgium
  4. 4.Department of MathematicsUniversity of Wisconsin–MadisonMadisonUSA

Personalised recommendations