On the Computational Content of the Brouwer Fixed Point Theorem
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.
Unable to display preview. Download preview PDF.
- 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
- 7.Hertling, P.: Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Informatik Berichte 208, FernUniversität Hagen, Hagen (November 1996)Google Scholar
- 9.Ishihara, H.: Reverse mathematics in Bishop’s constructive mathematics. Philosophia Scientiae, Cahier special 6, 43–59 (2006)Google Scholar
- 11.Miller, J.S.: Pi-0-1 Classes in Computable Analysis and Topology. Ph.D. thesis, Cornell University, Ithaca, USA (2002)Google Scholar
- 18.von Stein, T.: Vergleich nicht konstruktiv lösbarer Probleme in der Analysis. Diplomarbeit, Fachbereich Informatik, FernUniversität Hagen (1989)Google Scholar
- 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.Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen (September 1992)Google Scholar