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Conference on Computability in Europe

CiE 2013: The Nature of Computation. Logic, Algorithms, Applications pp 294–305Cite as

Closed Choice for Finite and for Convex Sets

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7921))

Abstract

We investigate choice principles in the Weihrauch lattice for finite sets on the one hand, and convex sets on the other hand. Increasing cardinality and increasing dimension both correspond to increasing Weihrauch degrees. Moreover, we demonstrate that the dimension of convex sets can be characterized by the cardinality of finite sets encodable into them. Precisely, choice from an n + 1 point set is reducible to choice from a convex set of dimension n, but not reducible to choice from a convex set of dimension n − 1.

This work benefited from the Royal Society International Exchange Grant IE111233 and the Marie Curie International Research Staff Exchange Scheme Computable Analysis, PIRSES-GA-2011- 294962.

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References

  1. Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Annals of Pure and Applied Logic 163(8), 968–1008 (2012)

    Article  Google Scholar 

  2. Brattka, V., Gherardi, G.: Borel complexity of topological operations on computable metric spaces. Journal of Logic and Computation 19(1), 45–76 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic 17(1), 73–117 (2011), arXiv:0905.4685

    Article  MathSciNet  MATH  Google Scholar 

  4. Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. Journal of Symbolic Logic 76, 143–176 (2011), arXiv:0905.4679

    Article  MathSciNet  MATH  Google Scholar 

  5. Brattka, V., Gherardi, G., Marcone, A.: The Bolzano-Weierstrass Theorem is the jump of Weak König’s Lemma. Annals of Pure and Applied Logic 163(6), 623–625 (2012), also arXiv:1101.0792

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

  7. Brattka, V., Pauly, A.: Computation with advice. Electronic Proceedings in Theoretical Computer Science 24 (2010), http://arxiv.org/html/1006.0551 cCA 2010

  8. Gasarch, W., Martin, G.: Bounded Queries in Recursion Theory. In: Progress in Computer Science & Applied Logic, vol. 16. Birkhäuser (1999)

    Google Scholar 

  9. Gherardi, G., Marcone, A.: How incomputable is the separable Hahn-Banach theorem? Notre Dame Journal of Formal Logic 50(4), 393–425 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hertling, P.: Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Ph.D. thesis, Fernuniversität, Gesamthochschule in Hagen (Oktober 1996)

    Google Scholar 

  11. Higuchi, K., Pauly, A.: The degree-structure of Weihrauch-reducibility. Logical Methods in Computer Science 9(2) (2013)

    Google Scholar 

  12. Kihara, T.: Incomputability of simply connected planar continua. Computability (2), 131–152 (2012)

    Google Scholar 

  13. Le Roux, S., Pauly, A.: Closed choice: Cardinality vs convex dimension. arXiv (2013)

    Google Scholar 

  14. Le Roux, S., Ziegler, M.: Singular coverings and non-uniform notions of closed set computability. Mathematical Logic Quarterly 54(5), 545–560 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Longpré, L., Kreinovich, V., Gasarch, W., Walster, W.: m solutions good, m − 1 solutions better. Applied Mathematical Sciences 2, 223–239 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Mylatz, U.: Vergleich unstetiger Funktionen in der Analysis. Master’s thesis, Fachbereich Informatik, FernUniversität Hagen (May 1992)

    Google Scholar 

  17. Mylatz, U.: Vergleich unstetiger Funktionen : “Principle of Omniscience” und Vollständigkeit in der C-Hierarchie. Ph.D. thesis, Fernuniversität, Gesamthochschule in Hagen (May 2006)

    Google Scholar 

  18. Pauly, A.: How incomputable is finding Nash equilibria? Journal of Universal Computer Science 16(18), 2686–2710 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Pauly, A.: On the (semi)lattices induced by continuous reducibilities. Mathematical Logic Quarterly 56(5), 488–502 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pauly, A., de Brecht, M.: Non-deterministic computation and the Jayne Rogers theorem. Electronic Proceedings in Theoretical Computer Science, DCM 2012 (2012) (to appear)

    Google Scholar 

  21. Stein, T.v.: Vergleich nicht konstruktiv lösbarer Probleme in der Analysis. Master’s thesis, Fachbereich Informatik, FernUniversität Hagen (1989)

    Google Scholar 

  22. Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Informatik Berichte 129, FernUniversität Hagen, Hagen (July 1992)

    Google Scholar 

  23. Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen (September 1992)

    Google Scholar 

  24. Weihrauch, K.: Computable Analysis. Springer (2000)

    Google Scholar 

  25. Ziegler, M.: Real hypercomputation and continuity. Theory of Computing Systems 41, 177–206 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Technische Universität Darmstadt, Germany

    Stéphane Le Roux

  2. Clare College, University of Cambridge, United Kingdom

    Arno Pauly

Authors
  1. Stéphane Le Roux
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  2. Arno Pauly
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Editor information

Editors and Affiliations

  1. Dipartimento die Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca, Piazza dell’Ateneo Nuovo 1, 20126, Milan, Italy

    Paola Bonizzoni

  2. Institute for Theoretical Computer Science, Mathematics and Operations Research, Faculty of Computer Science, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

    Vasco Brattka

  3. Institute for Logic, Language and Computation, University of Amsterdam, Postbus 94242, 1090 GE, Amsterdam, The Netherlands

    Benedikt Löwe

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Le Roux, S., Pauly, A. (2013). Closed Choice for Finite and for Convex Sets. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_34

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  • DOI: https://doi.org/10.1007/978-3-642-39053-1_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39052-4

  • Online ISBN: 978-3-642-39053-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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