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Representations of Analytic Functions and Weihrauch Degrees

  • Arno PaulyEmail author
  • Florian Steinberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

Abstract

This paper considers several representations of the analytic functions on the unit disk and their mutual translations. All translations that are not already computable are shown to be Weihrauch equivalent to closed choice on the natural numbers. Subsequently some similar considerations are carried out for representations of polynomials. In this case in addition to closed choice the Weihrauch degree \(\text {LPO}^*\) shows up as the difficulty of finding the degree or the zeros.

Keywords

Computable analysis Analytic function Weihrauch reduction Polynomials Closed choice LPO* 

Notes

Acknowledgements

The work has benefited from the Marie Curie International Research Staff Exchange Scheme Computable Analysis, PIRSES-GA-2011- 294962. The first author was supported partially by the ERC inVEST (279499) project, the second by the International Research Training Group 1529 ‘Mathematical Fluid Dynamics’ funded by the DFG and JSPS.

References

  1. 1.
    Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Ann. Pure Appl. Logic 163(8), 986–1008 (2012). http://dx.doi.org/10.1016/j.apal.2011.12.020 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symbolic Logic 17(1), 73–117 (2011). http://dx.doi.org/10.2178/bsl/1294186663 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brattka, V., Gherardi, G., Hölzl, R.: Probabilistic computability and choice. Inform. Comput. 242, 249–286 (2015). http://dx.doi.org/10.1016/j.ic.2015.03.005 CrossRefGoogle Scholar
  4. 4.
    Kawamura, A., Müller, N., Rösnick, C., Ziegler, M.: Computational benefit of smoothness: parameterized bit-complexity of numerical operators on analytic functions and Gevrey’s hierarchy. J. Complex. 31(5), 689–714 (2015). http://dx.doi.org/10.1016/j.jco.2015.05.001 CrossRefzbMATHGoogle Scholar
  5. 5.
    Ko, K.: Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser Boston Inc., Boston (1991). http://dx.doi.org/10.1007/978-1-4684-6802-1
  6. 6.
    Ko, K.: Polynomial-time computability in analysis. In: Handbook of Recursive Mathematics. Stud. Logic Found. Math., vols. 2 and 139, pp. 1271–1317. North-Holland, Amsterdam (1998). http://dx.doi.org/10.1016/S0049-237X(98)80052-9
  7. 7.
    Kunkle, D., Schröder, M.: Some examples of non-metrizable spaces allowing a simple type-2 complexity theory. In: Proceedings of the 6th Workshop on Computability and Complexity in Analysis (CCA 2004). Electron. Notes Theor. Comput. Sci., vol. 120, pp. 111–123. Elsevier, Amsterdam (2005). http://dx.doi.org/10.1016/j.entcs.2004.06.038
  8. 8.
    Le Roux, S., Pauly, A.: Finite choice, convex choice and finding roots. Logical Methods Comput. Sci. (2015, to appear). http://arxiv.org/abs/1302.0380
  9. 9.
    Müller, N.T.: Constructive aspects of analytic functions. In: Ko, K.I., Weihrauch, K. (eds.) Computability and Complexity in Analysis, CCA Workshop. Informatik Berichte, vol. 190, pp. 105–114. FernUniversität Hagen, Hagen (1995)Google Scholar
  10. 10.
    Mylatz, U.: Vergleich unstetiger funktionen in der analysis. Diplomarbeit, Fachbereich Informatik, FernUniversität Hagen (1992)Google Scholar
  11. 11.
    Mylatz, U.: Vergleich unstetiger funktionen: “Principle of Omniscience” und Vollständigkeit in der C-Hierarchie. Ph.D. thesis, Fernuniversität, Gesamthochschule in Hagen (Mai 2006)Google Scholar
  12. 12.
    Pauly, A.: Methoden zum Vergleich der Unstetigkeit von Funktionen. Masters thesis, FernUniversität Hagen (2007)Google Scholar
  13. 13.
    Pauly, A.: Infinite oracle queries in type-2 machines (extended abstract). arXiv:0907.3230v1, July 2009
  14. 14.
    Pauly, A.: On the topological aspects of the theory of represented spaces. Computability (201X, accepted for publication). http://arxiv.org/abs/1204.3763
  15. 15.
    Pauly, A., de Brecht, M.: Non-deterministic computation and the Jayne Rogers theorem. In: DCM 2012. Electronic Proceedings in Theoretical Computer Science, vol. 143 (2014)Google Scholar
  16. 16.
    Pauly, A., Davie, G., Fouché, W.: Weihrauch-completeness for layerwise computability. arXiv:1505.02091 (2015)
  17. 17.
    Pauly, A., Steinberg, F.: Representations of analytic functions and Weihrauch degrees. arXiv:1512.03024
  18. 18.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)Google Scholar
  19. 19.
    Weihrauch, K.: Computable Analysis. An Introduction. Texts in Theoretical Computer Science, An EATCS Series. Springer, Berlin (2000). http://dx.doi.org/10.1007/978-3-642-56999-9
  20. 20.
    Ziegler, M.: Real computation with least discrete advice: a complexity theory of nonuniform computability with applications to effective linear algebra. Ann. Pure Appl. Logic 163(8), 1108–1139 (2012). http://dx.doi.org/10.1016/j.apal.2011.12.030 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Computer LaboratoryUniversity of CambridgeCambridgeUK
  2. 2.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  3. 3.Fachbereich MathematikTU-DarmstadtDarmstadtGermany

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