Theory of Computing Systems

, Volume 60, Issue 3, pp 396–420 | Cite as

Genericity of Weakly Computable Objects

  • Mathieu Hoyrup


In computability theory many results state the existence of objects that in many respects lack algorithmic structure but at the same time are effective in some sense. Friedberg and Muchnik’s answer to Post’s problem is one of the most celebrated results in this form. The main goal of the paper is to develop a general result that embodies a large number of these particular constructions, capturing the essential idea that is common to all of them, and expressing it in topological terms. To do so, we introduce the effective topological notions of irreversible function and directional genericity and provide two main results that identify situations when such constructions are possible, clarifying the role of topology in many arguments from computability theory. We apply these abstract results to particular situations, illustrating their strength and deriving new results. This paper is an extended version of the conference paper (Hoyrup 6) with detailed proofs and new results.


Computable Generic Irreversible function Priority method 



The author wishes to thank Peter Gács, Emmanuel Jeandel and Cristóbal Rojas for discussions on the subject and the anonymous referees for very useful comments that helped improving the readability of the paper.


  1. 1.
    Brattka, V., Gherardi, G.: Weihrauch degrees omniscience principles and weak computability. J. Symb. Log. 76(1), 143–176 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brattka, V.: Computable versions of Baire’s category theorem. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS, pp 224–235. Springer-Verlag, London (2001)Google Scholar
  3. 3.
    Downey, R.G., Hirschfeldt, D.R., LaForte, G.: Randomness and reducibility. J. Comput. Syst. Sci. 68(1), 96–114 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Downey, R.G., LaForte, G.: Presentations of computably enumerable reals. Theor Comput. Sci. 284(2), 539–555 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hoyrup, M.: Randomness and the ergodic decomposition. In: Löwe, B., Normann, D., Soskov, I.N., Soskova, A.A. (eds.) CiE, volume 6735 of Lecture Notes in Computer Science, pp 122–131. Springer (2011)Google Scholar
  6. 6.
    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, March 5-8, 2014, Lyon, France, volume 25 of LIPIcs, pp 362–373. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)Google Scholar
  7. 7.
    Hocking, J.G., Young, G.S.: Topology. Addison-Wesley series in mathematics. Dover Publications (1961)Google Scholar
  8. 8.
    Ingrassia, M.A.: P-Genericity for Recursively Enumerable Sets. PhD thesis. University of Illinois at Urbana-Champaign (1981)Google Scholar
  9. 9.
    Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71, 152-182 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Miller, J.S.: Degrees of unsolvability of continuous functions. J. Symb. Log. 69(2), 555–584 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nies, A.: Computability and randomness. Oxford logic guides. Oxford University Press (2009)Google Scholar
  12. 12.
    Parthasarathy, K.R.: On the category of ergodic measures. Ill. J. Math. 5, 648–656 (1961)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Phelps, R.R.: Lectures on Choquet’s Theorem. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Selman, A.L.: Arithmetical reducibilities I. Math. Log. Q. 17(1), 335–350 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stephan, F., Wu, G.: Presentations of K-trivial reals and Ksolmogorov complexity. In: CiE, pp 461–469. Springer-Verlag, Berlin, Heidelberg (2005)Google Scholar
  16. 16.
    V’yugin, V.V.: Effective Convergence in probability and an ergodic theorem for individual random sequences. SIAM Theory Probab. Appl. 42(1), 39–50 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen (1992)Google Scholar
  18. 18.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  19. 19.
    Yasugi, M., Mori, T., Tsujii, Y.: Effective properties of sets and functions in metric spaces with computability structure. Theor. Comput. Sci. 219(1-2), 467–486 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.InriaVillers-lès-NancyFrance

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