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Effective Choice and Boundedness Principles in Computable Analysis

  • Vasco Brattka (a1) and Guido Gherardi (a2)
Abstract

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn–Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example.

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[1] Bishop, Errett and Bridges, Douglas S., Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer, Berlin, 1985.
[2] Brattka, Vasco, Computable invariance, Theoretical Computer Science, vol. 210 (1999), pp. 320.
[3] Brattka, Vasco, Computable versions of Baire's category theorem, 26th International symposium, Mathematical Foundations of Computer Science, MFCS 2001 (Sgall, Jičí, Pultr, Aleš, and Kolman, Petr, editors), Lecture Notes in Computer Science, vol. 2136, Springer, Berlin, 2001, pp. 224235.
[4] Brattka, Vasco, Effective representations of the space of linear bounded operators, Applied General Topology, vol. 4 (2003), no. 1, pp. 115131.
[5] Brattka, Vasco, Effective Borel measurability and reducibility of functions, Mathematical Logic Quarterly, vol. 51 (2005), no. 1, pp. 1944.
[6] Brattka, Vasco, Computable versions of the uniform boundedness theorem, Logic colloquium 2002 (Chatzidakis, Z., Koepke, P., and Pohlers, W., editors), Lecture Notes in Logic, vol. 27, Association for Symbolic Logic, 2006, pp. 130151.
[7] Brattka, Vasco, Borel complexity and computability of the Hahn–Banach Theorem, Archive for Mathematical Logic, vol. 46 (2008), no. 7–8, pp. 547564.
[8] Brattka, Vasco, Plottable real number functions and the computable graph theorem, SIAM Journal on Computing, vol. 38 (2008), no. 1, pp. 303328.
[9] Brattka, Vasco, A computable version of Banach's inverse mapping theorem, Annals of Pure and Applied Logic, vol. 157 (2009), pp. 8596.
[10] Brattka, Vasco and Gherardi, Guido, Borel complexity of topological operations on computable metric spaces, Journal of Logic and Computation, vol. 19 (2009), no. 1, pp. 4576.
[11] Brattka, Vasco and Gherardi, Guido, Effective choice and boundedness principles in computable analysis, Proceedings of the sixth international conference on Computability and Complexity in Analysis, CCA 2009 (Bauer, Andrej, Hertling, Peter, and Ko, Ker-I, editors), Leibniz–Zentrum für Informatik, Schloss Dagstuhl, Germany, 2009, pp. 95106.
[12] Brattka, Vasco and Gherardi, Guido, Weihrauch degrees, omniscience principles and weak computability, The Journal of Symbolic Logic, vol. 76 (2011), no. 1, pp. 143176.
[13] Brattka, Vasco and Presser, Gero, Computability on subsets of metric spaces, Theoretical Computer Science, vol. 305 (2003), pp. 4376.
[14] Brattka, Vasco and Weihrauch, Klaus, Computability on subsets of Euclidean space I: Closed and compact subsets, Theoretical Computer Science, vol. 219 (1999), pp. 6593.
[15] Bridges, Douglas and Richman, Fred, Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987.
[16] Caldwell, J. and Pour-El, Marian Boykan, On a simple definition of computable functions of a real variable—with applications to functions of a complex variable, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 119.
[17] Cenzer, Douglas and Jockusch, Carl G. Jr., classes—structure and applications, Computability theory and its applications, Contemporary Mathematics, vol. 257, American Mathematical Society, Providence, RI, 2000, pp. 3959.
[18] Gelfond, Michael G., A class of theorems with valid constructive counterparts, Constructive mathematics (Richman, Fred, editor), Lecture Notes in Mathematics, vol. 873, Springer, Berlin, 1981, pp. 314320.
[19] Gherardi, Guido, Effective Borel degrees of some topological functions, Mathematical Logic Quarterly, vol. 52 (2006), no. 6, pp. 625642.
[20] Gherardi, Guido and Marcone, Alberto, How incomputable is the separable Hahn–Banach theorem?, Notre Dame Journal of Formal Logic, vol. 50 (2009), pp. 393425.
[21] Hauck, Jürgen, Berechenbare reelle Funktionenfolgen, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 265282.
[22] Hayashi, Susumu, Mathematics based on incremental learning–excluded middle and inductive inference, Theoretical Computer Science, vol. 350 (2006), pp. 125139.
[23] Hertling, Peter, Unstetigkeitsgrade von Funktionen in der effektiven Analysis, Dissertation, Fern Universität Hagen, Hagen, November 1996.
[24] Ishihara, Hajime, An omniscience principle, the König lemma and the Hahn–Banach theorem, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 36 (1990), pp. 237240.
[25] Ishihara, Hajime, Informal constructive reverse mathematics, CDMTCS research report, Centre for Discrete Mathematics and Theoretical Computer Science, 2004.
[26] Ishihara, Hajime, Constructive reverse mathematics: compactness properties, From sets and types to topology and analysis: Towards practicable foundations for constructive mathematics (Crosilla, Laura and Schuster, Peter, editors), Oxford University Press, 2005, pp. 245267.
[27] Ishihara, Hajime, Unique existence and computability in constructive reverse mathematics, Computation and logic in the real world, third conference on Computability in Europe, CiE 2007 (Cooper, S. Barry, Löwe, Benedikt, and Sorbi, Andrea, editors), Lecture Notes in Computer Science, vol. 4497, Springer, Berlin, 2007, pp. 368377.
[28] Jockusch, Carl G. Jr. and Soare, Robert I., Degrees of members of classes, Pacific Journal of Mathematics, vol. 40 (1972), pp. 605616.
[29] Kleene, S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 4173.
[30] Kohlenbach, Ulrich, The use of a logical principle of uniform boundedness in analysis, Logic and foundations of mathematics (Cantini, A., Casari, E., and Minari, P., editors), Synthese Library, vol. 280, Kluwer Academic Publishers, 1999, pp. 93106.
[31] Kohlenbach, Ulrich, On uniform weak König's lemma, Annals of Pure and Applied Logic, vol. 114 (2002), pp. 103116.
[32] Kohlenbach, Ulrich, Higher order reverse mathematics, Reverse mathematics 2001 (Simpson, Stephen G., editor), Lecture Notes in Logic, vol. 21, A K Peters, 2005, pp. 281295.
[33] Kreisel, Georg and Lacombe, Daniel, Ensembles récursivement mesurables et ensembles récursivement ouverts et fermés, Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 245 (1957), pp. 11061109.
[34] Mandelkern, Mark, Constructively complete finite sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), no. 2, pp. 97103.
[35] Mylatz, Uwe, Vergleich unstetiger Funktionen: “Principle of Omniscience” und Vollständigkeit in der C–hierarchie, Ph.D. thesis, Faculty for Mathematics and Computer Science, University Hagen, Germany, 2006.
[36] Nakata, Masahiro and Hayashi, Susumu, A limiting first order realizability interpretation, Scientiae Mathematicae Japonicae, vol. 55 (2002), no. 3, pp. 567580.
[37] Odifreddi, Piergiorgio, Classical recursion theory, Studies in Logic and the Foundations of Mathematics, vol. 125, North-Holland, Amsterdam, 1989.
[38] Pauly, Arno, On the (semi)lattices induced by continuous reducibilities, Mathematical Logic Quarterly, vol. 56 (2010), no. 5, pp. 488502.
[39] Pour-El, Marian B. and Richards, J. Ian, Computability in analysis and physics, Perspectives in Mathematical Logic, Springer, Berlin, 1989.
[40] Richman, Fred, Polynomials and linear transformations, Linear Algebra and its Applications, vol. 131 (1990), no. 1, pp. 131137.
[41] Richman, Fred, Omniscience principles and functions of bounded variation, Mathematical Logic Quarterly, vol. 48 (2002), no. 1, pp. 111116.
[42] Rogers, Hartley, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[43] Sakamoto, Nobuyuki and Yamazaki, Takeshi, Uniform versions of some axioms of second order arithmetic, Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 587593.
[44] Schuster, Peter, Unique solutions, Mathematical Logic Quarterly, vol. 52 (2006), no. 6, pp. 534539.
[45] Schuster, Peter, Corrigendum to “unique solutions”, Mathematical Logic Quarterly, vol. 53 (2007), no. 2, p. 214.
[46] Simpson, Stephen, Tanaka, Kazuyuki, and Yamazaki, Takeshi, Some conservation results on weak König's lemma, Annals of Pure and Applied Logic, vol. 118 (2002), no. 1–2, pp. 87114.
[47] Simpson, Stephen G., Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer, Berlin, 1999.
[48] Weidmann, Joachim, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer, Berlin, 1980.
[49] Weihrauch, Klaus, 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.
[50] Weihrauch, Klaus, The TTE-interpretation of three hierarchies of omniscience principles, Informatik Berichte 130, FernUniversität Hagen, Hagen, September 1992.
[51] Weihrauch, Klaus, Computable analysis, Springer, Berlin, 2000.
[52] Weihrauch, Klaus, On computable metric spaces Tietze–Urysohn extension is computable, 4th International workshop on Computability and Complexity in Analysis, CCA 2000 (Blanck, Jens, Brattka, Vasco, and Hertling, Peter, editors), Lecture Notes in Computer Science, vol. 2064, Springer, Berlin, 2001, pp. 357368.
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Bulletin of Symbolic Logic
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