Advertisement

Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 665–678 | Cite as

Schnorr trivial reals: a construction

  • Johanna N. Y. Franklin
Article

Abstract

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null \({\Sigma^0_1}\) (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if \({{\bf h'} \geq_T {\bf 0''}}\) , then h contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees.

Keywords

Randomness Triviality Schnorr trivial 

Mathematical Subject Classification (2000)

03D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chaitin, G.J.: A theory of program size formally identical to information theory. J. Assoc. Comput. Mach. 22, 329–340 (1975)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chaitin, G.J.: Algorithmic information theory. IBM J. Res. Develop. 21(4), 350–359 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Downey, R., Griffiths, E., Laforte, G.: On Schnorr and computable randomness, martingales, and machines. Math. Log. Q. 50(6), 613–627 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Downey, R., Hirschfeldt, D.R., Nies, A., Terwijn, S.A.: Calibrating randomness. Bull. Symb. Logic 12(3), 411–491 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Downey, R.G., Griffiths, E.J.: Schnorr randomness. J. Symb. Logic 69(2), 533–554 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Downey, R.G., Hirschfeldt, D.R., Nies, A., Stephan, F.: Trivial reals. In: Proceedings of the 7th and 8th Asian Logic Conferences, pp. 103–131. Singapore University Press, Singapore (2003)Google Scholar
  7. 7.
    Franklin, J.N.Y.: Hyperimmune-free degrees and Schnorr triviality. SubmittedGoogle Scholar
  8. 8.
    Kjos-Hanssen, B., Nies, A., Stephan, F.: Lowness for the class of Schnorr random reals. SIAM J. Comput. 35(3), 647–657 (2005) (electronic)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Kučera, A.: Measure, \({\Pi^{0}_{1}}\) -classes and complete extensions of PA. In: Recursion theory week (Oberwolfach, 1984). Lecture Notes in Mathematics, vol. 1141, pp. 245–259. Springer, Berlin (1985)Google Scholar
  10. 10.
    Martin-Löf, P.: The definition of random sequences. Inform. Control 9, 602–619 (1966)CrossRefGoogle Scholar
  11. 11.
    Miller, W., Martin, D.A.: The degrees of hyperimmune sets. Z. Math. Logik Grundlagen Math. 14, 159–166 (1968)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Nies, A.: Lowness properties and randomness. Adv. Math. 197(1), 274–305 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Nies, A., Stephan, F., Terwijn, S.A.: Randomness, relativization and Turing degrees. J. Symb. Logic 70(2), 515–535 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Schnorr, C.P.: A unified approach to the definition of random sequences. Math. Syst. Theory 5, 246–258 (1971)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Terwijn, S.A., Zambella, D.: Computational randomness and lowness. J. Symb. Logic 66(3), 1199–1205 (2001)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations