Notre Dame Journal of Formal Logic

How Incomputable Is the Separable Hahn-Banach Theorem?

Guido Gherardi and Alberto Marcone


We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

Article information

Notre Dame J. Formal Logic Volume 50, Number 4 (2009), 393 - 425 .

First available in Project Euclid: 11 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F60
Secondary: 03B30 46A22 46S30

computable analysis reverse mathematics weak Konig's lemma Hahn-Banach extension theorem multivalued functions


Gherardi , Guido; Marcone , Alberto. How Incomputable Is the Separable Hahn-Banach Theorem?. Notre Dame J. Formal Logic 50 (2009), no. 4, 393 -- 425. doi:10.1215/00294527-2009-018.

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  • [1] Brattka, V., "Effective Borel measurability and reducibility of functions", Mathematical Logic Quarterly, vol. 51 (2005), pp. 19--44.
  • [2] Brattka, V., "Borel complexity and computability of the Hahn-Banach Theorem", Archive for Mathematical Logic, vol. 46 (2008), pp. 547--64.
  • [3] Brattka, V., and G. Gherardi, "Borel complexity of topological operations on computable metric spaces", pp. 83--97 in Computation and Logic in the Real World. Proceedings of the Third Conference on Computability in Europe, CiE 2007, Siena, 2007, vol. 4497 of Lecture Notes in Computer Science, Springer, Berlin, 2007.
  • [4] Brattka, V., and G. Gherardi, "Borel complexity of topological operations on computable metric spaces", Journal of Logic and Computation, vol. 19 (2009), pp. 45--76.
  • [5] Brattka, V., and G. Presser, "Computability on subsets of metric spaces", Theoretical Computer Science, vol. 305 (2003), pp. 43--76. Topology in Computer Science (Schloß Dagstuhl, 2000).
  • [6] Brown, D. K., and S. G. Simpson, ``Which set existence axioms are needed to prove the separable Hahn-Banach theorem?'' Annals of Pure and Applied Logic, vol. 31 (1986), pp. 123--44. Special issue: Second Southeast Asian logic conference (Bangkok, 1984).
  • [7] Friedman, H., "Some systems of second order arithmetic and their use", pp. 235--42 in Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 1, Canadian Mathematics Congress, Montreal, 1975.
  • [8] Grubba, T., M. Schröder, and K. Weihrauch, "Computable metrization", Mathematical Logic Quarterly, vol. 53 (2007), pp. 381--95.
  • [9] Grubba, T., and K. Weihrauch, "A computable version of Dini's theorem for topological spaces", pp. 117--29 in Computability and Complexity in Analysis, edited by T. Grubba, P. Hertling, H. Tsuiki, and K. Weihrauch, vol. 326 of Informatik Berichte, FernUniversität Hagen, 2005.
  • [10] Grzegorczyk, A., "Computable functionals", Fundamenta Mathematicae, vol. 42 (1955), pp. 168--202.
  • [11] Humphreys, A. J., and S. G. Simpson, "Separation and weak König's lemma", The Journal of Symbolic Logic, vol. 64 (1999), pp. 268--78.
  • [12] Kechris, A. S., Classical Descriptive Set Theory, Number 156 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
  • [13] Lacombe, D., "Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles. I", Comptes Rendus de l'Acadèmie des Sciences Paris, vol. 240 (1955), pp. 2478--80.
  • [14] Metakides, G., A. Nerode, and R. A. Shore, "Recursive limits on the Hahn-Banach theorem", pp. 85--91 in Errett Bishop: Reflections on Him and His Research (San Diego, 1983), vol. 39 of Contemporary Mathematics, American Mathematical Society, Providence, 1985.
  • [15] Mylatz, U., Vergleich unstetiger Funktionen in der Analysis, Ph.D. thesis, FernUniversität Hagen, 1989.
  • [16] Pour-El, M. B., and J. I. Richards, Computability in Analysis and Physics, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1989.
  • [17] Shioji, N., and K. Tanaka, "Fixed point theory in weak second-order arithmetic", Annals of Pure and Applied Logic, vol. 47 (1990), pp. 167--88.
  • [18] Simpson, S. G., editor, Reverse Mathematics 2001, Lecture Notes in Logic. Association for Symbolic Logic, La Jolla, 2005.
  • [19] Simpson, S. G., Subsystems of Second Order Arithmetic, Springer-Verlag, Berlin, 1999.
  • [20] von Stein, T., Vergleich nicht konstruktiv lösbarer Probleme in der Analysis, Ph.D. thesis, FernUniversität Hagen, 1989.
  • [21] Weihrauch, K., Computable Analysis. An Introduction, Texts in Theoretical Computer Science. Springer-Verlag, Berlin, 2000.
  • [22] Weihrauch, K., "On computable metric spaces Tietze-Urysohn extension is computable", pp. 357--68 in Computability and Complexity in Analysis (Swansea, 2000), vol. 2064 of Lecture Notes in Computer Science, Springer, Berlin, 2001.