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A comparison of concepts from computable analysis and effective descriptive set theory

Published online by Cambridge University Press:  23 June 2016

VASSILIOS GREGORIADES ,
TAMÁS KISPÉTER  and
ARNO PAULY
VASSILIOS GREGORIADES
Affiliation:
Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany Email: gregoriades@mathematik.tu-darmstadt.de
TAMÁS KISPÉTER
Affiliation:
Computer Laboratory, University of Cambridge, Cambridge, U.K. Email: tk407@cam.ac.uk, Arno.Pauly@cl.cam.ac.uk
ARNO PAULY
Affiliation:
Computer Laboratory, University of Cambridge, Cambridge, U.K. Email: tk407@cam.ac.uk, Arno.Pauly@cl.cam.ac.uk
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Abstract

Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.

As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 8: Continuity, Computability, Constructivity: From Logic to Algorithms 2013 , December 2017 , pp. 1414 - 1436
Copyright
Copyright © Cambridge University Press 2016 

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References

Brattka, V. (2005). Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly 51 19–44.CrossRefGoogle Scholar
Brattka, V., de Brecht, M. and Pauly, A. (2012a). Closed choice and a uniform low basis theorem. Annals of Pure and Applied Logic 163 968–1008.Google Scholar
Brattka, V. and Gherardi, G. (2011). Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic 1 73–117. ArXiv:0905.4685.Google Scholar
Brattka, V., Gherardi, G. and Marcone, A. (2012b). The Bolzano-Weierstrass theorem is the jump of weak König's lemma. Annals of Pure and Applied Logic 163 623–625. Also arXiv:1101.0792.Google Scholar
Bruckner, A.M., Bruckner, J.B. and Thomson, B.S. (1997). Real Analysis, Prentice-Hall.Google Scholar
de Brecht, M. (2013). Quasi-polish spaces. Annals of Pure and Applied Logic 164 354–381.Google Scholar
de Brecht, M. (2014). Levels of discontinuity, limit-computability, and jump operators. In: Brattka, V., Diener, H. and Spreen, S. (eds.) Logic, Computation, Hierarchies, de Gruyter, 79–108. ArXiv 1312.0697.Google Scholar
de Brecht, M. and Yamamoto, A. (2009). σ0 α - admissible representations (extended abstract). In: Bauer, A., Hertling, P. and Ko, K.-I. (eds.) Proceedings of the 6th International Conference on Computability and Complexity in Analysis. Schloss Dagstuhl, http://drops.dagstuhl.de/opus/volltexte/2009/2264.Google Scholar
Gherardi, G. and Marcone, A. (2009). How incomputable is the separable Hahn-Banach theorem? Notre Dame Journal of Formal Logic 50 393–425.Google Scholar
Gregoriades, V. and Moschovakis, Y.N. (in preparation). Notes on effective descriptive set theory. Notes in preparation.Google Scholar
Hertling, P. (1996). Unstetigkeitsgrade von Funktionen in der effektiven Analysis, Ph.D. thesis, Fernuniversität, Gesamthochschule in Hagen.Google Scholar
Jayne, J.E. and Rogers, C.A. (1982). First level borel functions and isomorphisms. Journal de Mathématiques Pures et Appliqués 61 177–205.Google Scholar
Kačena, M., Ros, L.M. and Semmes, B. (2012). Some observations on ‘a new proof of a theorem of Jayne and Rogers’. Real Analysis Exchange 38 121–132.Google Scholar
Kihara, T. (2015). Decomposing Borel functions using the Shore-Slaman join theorem. Fundamenta Mathematicae 230 1–13. doi: 10.4064/fm230-1-1.Google Scholar
Korovina, M.V. and Kudinov, O.V. (2008). Towards computability over effectively enumerable topological spaces. Electronic Notes in Theoretical Computer Science 221 115–125.Google Scholar
Kreitz, C. and Weihrauch, K. (1985). Theory of representations. Theoretical Computer Science 38 35–53.Google Scholar
Moschovakis, Y.N. (1980). Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics, volume 100, North-Holland.Google Scholar
Moschovakis, Y.N. (2010). Classical descriptive set theory as a refinement of effective descriptive set theory. Annals of Pure and Applied Logic 162 243–255.CrossRefGoogle Scholar
Motto Ros, L., Schlicht, P. and Selivanov, V. (2014). Wadge-like reducibilities on arbitrary quasi-polish spaces. Mathematical Structures in Computer Science 1–50, http://journals.cambridge.org/article_S0960129513000339. ArXiv 1204.5338.Google Scholar
Motto Ros, L. and Semmes, B. (2009). A new proof of a theorem of Jayne and Rogers. Real Analysis Exchange 35 195–204.Google Scholar
Pauly, A. (2010). How incomputable is finding Nash equilibria? Journal of Universal Computer Science 16 2686–2710.Google Scholar
Pauly, A. (2012). On the topological aspects of the theory of represented spaces. http://arxiv.org/abs/1204.3763.Google Scholar
Pauly, A. (2015a). Computability on the countable ordinals and the Hausdorff-Kuratowski theorem. arXiv 1501.00386.Google Scholar
Pauly, A. (2015b). Computability on the countable ordinals and the hausdorff-kuratowski theorem (extended abstract). In: Italiano, G.F., Pighizzini, G. and Sannella, D.T. (eds.) Mathematical Foundations of Computer Science 2015. Springer Lecture Notes in Computer Science 9234 407–418, http://dx.doi.org/10.1007/978-3-662-48057-1_32.Google Scholar
Pauly, A. and de Brecht, M. (2013). Towards synthetic descriptive set theory: An instantiation with represented spaces. arXiv 1307.1850.Google Scholar
Pauly, A. and de Brecht, M. (2014). Non-deterministic computation and the Jayne Rogers theorem. Electronic Proceedings in Theoretical Computer Science 143 87–96.Google Scholar
Pauly, A. and de Brecht, M. (2015). Descriptive set theory in the category of represented spaces. In: Proceedings of the 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 438–449.Google Scholar
Pour-El, M. and Richards, I. (1989). Computability in Analysis and Physics, Perspectives in Mathematical Logic, Springer.CrossRefGoogle Scholar
Rettinger, R. and Weihrauch, K. (2013). Products of effective topological spaces and a uniformly computable tychonoff theorem. Logical Methods in Computer Science 9.Google Scholar
Saint Raymond, J. (2007). Preservation of the borel class under countable-compact-covering mappings. Topology and its Applications 154 1714–1725.Google Scholar
Schröder, M. (2002). Admissible Representations for Continuous Computations, Ph.D. Thesis, FernUniversität Hagen.Google Scholar
Selivanov, V. L. (2013). Total representations. Logical Methods in Computer Science 9.Google Scholar
Semmes, B. (2009). A Game for the Borel Functions, Ph.D. Thesis, University of Amsterdam.Google Scholar
Weihrauch, K. (1987). Computability, Monographs on Theoretical Computer Science, Springer-Verlag.Google Scholar
Weihrauch, K. (1992a). The degrees of discontinuity of some translators between representations of the real numbers. In: Informatik Berichte, volume 129, FernUniversität Hagen, Hagen.Google Scholar
Weihrauch, K. (1992b). The TTE-interpretation of three hierarchies of omniscience principles. In: Informatik Berichte, volume 130, FernUniversität Hagen, Hagen.Google Scholar
Weihrauch, K. (2000). Computable Analysis, Springer-Verlag.Google Scholar
Ziegler, M. (2007). Real hypercomputation and continuity. Theory of Computing Systems 41 177–206.Google Scholar