On the Computational Content of the Brouwer Fixed Point Theorem

  • Vasco Brattka
  • Stéphane Le Roux
  • Arno Pauly
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)

Abstract

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. Another main result is that connected choice is complete for dimension greater or equal to three in the sense that it is computably equivalent to Weak Kőnig’s Lemma. In contrast to this, the connected choice operations in dimensions zero, one and two form a strictly increasing sequence of Weihrauch degrees, where connected choice of dimension one is known to be equivalent to the Intermediate Value Theorem. Whether connected choice of dimension two is strictly below connected choice of dimension three or equivalent to it is unknown, but we conjecture that the reduction is strict. As a side result we also prove that finding a connectedness component in a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baigger, G.: Die Nichtkonstruktivität des Brouwerschen Fixpunktsatzes. Arch. Math. Logik Grundlag. 25, 183–188 (1985)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a Uniform Low Basis Theorem. Annals of Pure and Applied Logic 163(8), 986–1008 (2012)Google Scholar
  3. 3.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. The Bulletin of Symbolic Logic 17(1), 73–117 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. The Journal of Symbolic Logic 76(1), 143–176 (2011)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brattka, V., Gherardi, G., Marcone, A.: The Bolzano-Weierstrass theorem is the jump of weak Kőnig’s lemma. Annals of Pure and Applied Logic 163, 623–655 (2012)MATHCrossRefGoogle Scholar
  6. 6.
    Gherardi, G., Marcone, A.: How incomputable is the separable Hahn-Banach theorem? Notre Dame Journal of Formal Logic 50, 393–425 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hertling, P.: Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Informatik Berichte 208, FernUniversität Hagen, Hagen (November 1996)Google Scholar
  8. 8.
    Hoyrup, M., Rojas, C., Weihrauch, K.: Computability of the Radon-Nikodym Derivative. In: Löwe, B., Normann, D., Soskov, I., Soskova, A. (eds.) CiE 2011. LNCS, vol. 6735, pp. 132–141. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Ishihara, H.: Reverse mathematics in Bishop’s constructive mathematics. Philosophia Scientiae, Cahier special 6, 43–59 (2006)Google Scholar
  10. 10.
    Le Roux, S., Ziegler, M.: Singular coverings and non-uniform notions of closed set computability. Mathematical Logic Quarterly 54(5), 545–560 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Miller, J.S.: Pi-0-1 Classes in Computable Analysis and Topology. Ph.D. thesis, Cornell University, Ithaca, USA (2002)Google Scholar
  12. 12.
    Orevkov, V.: 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. 4, 1253–1256 (1963)MathSciNetGoogle Scholar
  13. 13.
    Pauly, A.: How incomputable is finding Nash equilibria? Journal of Universal Computer Science 16(18), 2686–2710 (2010)MathSciNetMATHGoogle Scholar
  14. 14.
    Pauly, A.: On the (semi)lattices induced by continuous reducibilities. Mathematical Logic Quarterly 56(5), 488–502 (2010)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Richter, M.K., Wong, K.C.: Non-computability of competitive equilibrium. Economic Theory 14(1), 1–27 (1999)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Shioji, N., Tanaka, K.: Fixed point theory in weak second-order arithmetic. Annals of Pure and Applied Logic 47, 167–188 (1990)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1999)MATHCrossRefGoogle Scholar
  18. 18.
    von Stein, T.: Vergleich nicht konstruktiv lösbarer Probleme in der Analysis. Diplomarbeit, Fachbereich Informatik, FernUniversität Hagen (1989)Google Scholar
  19. 19.
    Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Technical Report TR-92-050, International Computer Science Institute, Berkeley (July 1992)Google Scholar
  20. 20.
    Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen (September 1992)Google Scholar
  21. 21.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Vasco Brattka
    • 1
  • Stéphane Le Roux
    • 2
  • Arno Pauly
    • 3
  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa
  2. 2.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Computer LaboratoryUniversity of CambridgeCambridgeUnited Kingdom

Personalised recommendations