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Conference on Computability in Europe

CiE 2016: Pursuit of the Universal pp 115–123Cite as

The Typical Constructible Object

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9709))

Abstract

Baire Category is an important concept in mathematical analysis. It gives a notion of large set, hence a way of identifying the properties of typical objects. One of the most important applications of Baire Category is to provide a way of proving the existence of objects with specified properties without having to give an explicit construction, showing at the same time that these properties are prevalent. For instance it has been extensively used in mathematical analysis to better understand and separate classes of real functions such as analytic and smooth functions (see [9] for a wide range of applications of the Baire Category Theorem in analysis).

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References

  1. Brattka, V., Hendtlass, M., Kreuzer, A.P.: On the uniform computational content of the Baire category theorem (2015). CoRR arXiv:1510.01913

  2. Bridges, D., Ishihara, H., Vîă, L.: A new constructive version of Baire’s theorem. Hokkaido Math. J. 35(1), 107–118 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brown, D.K., Simpson, S.G.: The Baire category theorem in weak subsystems of second-order arithmetic. J. Symb. Logic 58(2), 557–578 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Richard, M.: Friedberg: two recursively enumerable sets of incomparable degrees of unsolvability (solution of Post’s problem, 1944). Proc. Natl. Acad. Sci. U.S.A. 43(2), 236–238 (1957)

    Article  MATH  Google Scholar 

  5. Hoyrup, M.: Irreversible computable functions. In: Mayr, E.W., Portier, N. (eds.) 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), STACS 2014, 5–8 March 2014, Lyon, France, LIPIcs, vol. 25, pp. 362–373. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)

    Google Scholar 

  6. Hoyrup, M.: Genericity of weakly computable objects. Theory of Computing Systems (2016, to appear)

    Google Scholar 

  7. Ingrassia, M.A.: P-genericity for recursively enumerable sets. Ph.D. thesis, University of Illinois at Urbana-Champaign (1981)

    Google Scholar 

  8. Carl, G.: Jockush: simple proofs of some theorems on high degrees. Can. J. Math. 29, 1072–1080 (1977)

    Article  MATH  Google Scholar 

  9. Jones, S.H.: Applications of the Baire category theorem. Real Anal. Exch. 23(2), 363–394 (1999)

    MathSciNet  MATH  Google Scholar 

  10. Kleene, S.C., Emil, L.: Post: the upper semi-lattice of degrees of recursive unsolvability. Ann. Math. 59(3), 379–407 (1954)

    Article  MATH  Google Scholar 

  11. Jack, H.: Lutz: category and measure in complexity classes. SIAM J. Comput. 19(6), 1100–1131 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lutz, J.H.: Effective fractal dimensions. Mathe. Logic Q. 51(1), 62–72 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Muchnik, A.A.: On the unsolvability of the problem of reducibility in the theory of algorithms. Dokl. Akad. Nauk SSSR 108, 194–197 (1956)

    MathSciNet  MATH  Google Scholar 

  14. Nies, A.: Computability and Randomness. Oxford Logic Guides. Oxford University Press, Oxford (2009)

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. LORIA, Inria Nancy Grand Est, Villers-lès-Nancy, France

    Mathieu Hoyrup

Authors
  1. Mathieu Hoyrup
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Correspondence to Mathieu Hoyrup .

Editor information

Editors and Affiliations

  1. Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. Université Paris-Diderot - Paris 7, Paris, France

    Laurent Bienvenu

  3. University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

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© 2016 Springer International Publishing Switzerland

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Hoyrup, M. (2016). The Typical Constructible Object. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_12

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  • DOI: https://doi.org/10.1007/978-3-319-40189-8_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40188-1

  • Online ISBN: 978-3-319-40189-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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