Theory of Computing Systems

, Volume 56, Issue 3, pp 465–486 | Cite as

Schnorr Triviality and Its Equivalent Notions



We give some characterizations of Schnorr triviality. In concrete terms, we introduce a reducibility related to decidable prefix-free machines and show the equivalence with Schnorr reducibility. We also give a uniform-Schnorr-randomness version of the equivalence of LR-reducibility and LK-reducibility. Finally we prove a base-type characterization of Schnorr triviality.


Algorithmic randomness Schnorr randomness Schnorr triviality Uniform relativization Lowness for uniform Schnorr randomness 


  1. 1.
    Bienvenu, L., Merkle, W.: Reconciling data compression and Kolmogorov complexity. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 4596, pp. 643–654. Springer, Berlin (2007) CrossRefGoogle Scholar
  2. 2.
    Bienvenu, L., Miller, J.S.: Randomness and lowness notions via open covers. Ann. Pure Appl. Log. 163, 506–518 (2012). arXiv:1303.4902 CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Brattka, V.: Computability over topological structures. In: Cooper, S.B., Goncharov, S.S. (eds.) Computability and Models, pp. 93–136. Kluwer Academic, New York (2003) CrossRefGoogle Scholar
  4. 4.
    Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 425–491. Springer, Berlin (2008) CrossRefGoogle Scholar
  5. 5.
    Diamondstone, D., Greenberg, N., Turetsky, D.: A van Lambalgen theorem for Demuth randomness. In: Downey, R., Brendle, J., Goldblatt, R., Kim, B. (eds.) Proceedings of the 12th Asian Logic Conference, pp. 115–124 (2013) CrossRefGoogle Scholar
  6. 6.
    Downey, R., Greenberg, N., Mihailovic, N., Nies, A.: Lowness for computable machines. In: Chong, C.T., Feng, Q., Slaman, T.A., Woodin, W.H., Yang, Y. (eds.) Computational Prospects of Infinity: Part II. Lecture Notes Series, pp. 79–86. World Scientific, Singapore (2008) CrossRefGoogle Scholar
  7. 7.
    Downey, R., Griffiths, E.: Schnorr randomness. J. Symb. Log. 69(2), 533–554 (2004) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Downey, R., Griffiths, E., LaForte, G.: On Schnorr and computable randomness, martingales, and machines. Math. Log. Q. 50(6), 613–627 (2004) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Downey, R., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, Berlin (2010) CrossRefMATHGoogle Scholar
  10. 10.
    Franklin, J.N.Y., Stephan, F.: Schnorr trivial sets and truth-table reducibility. J. Symb. Log. 75(2), 501–521 (2010) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Franklin, J.N.Y., Stephan, F., Yu, L.: Relativizations of randomness and genericity notions. Bull. Lond. Math. Soc. 43(4), 721–733 (2011) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hirschfeldt, D., Nies, A., Stephan, F.: Using random sets as oracles. J. Lond. Math. Soc. 75, 610–622 (2007) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Hölzl, R., Merkle, W.: Traceable sets. In: Calude, C.S., Sassone, V. (eds.) Proceedings of Theoretical Computer Science—6th IFIP TC 1/WG 2.2 International Conference, TCS 2010, Held as Part of WCC 2010, IFIP TCS, Brisbane, Australia, September 20–23. IFIP Advances in Information and Communication Technology, vol. 323, pp. 301–315. Springer, Berlin (2010). ISBN 978-3-642-15239-9 CrossRefGoogle Scholar
  14. 14.
    Kihara, T., Miyabe, K.: Uniform Kurtz randomness. Submitted Google Scholar
  15. 15.
    Kjos-Hanssen, B., Miller, J.S., Solomon, D.R.: Lowness notions, measure, and domination. J. Lond. Math. Soc. 85(3), 869–888 (2012) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Kjos-Hanssen, B., Nies, A., Stephan, F.: Lowness for the class of Schnorr random reals. SIAM J. Comput. 35(3), 647–657 (2005) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kučera, A.: Measure, \(\varPi^{0}_{1}\) classes, and complete extensions of PA. In: Recursion Theory Week. Lecture Notes in Mathematics, vol. 1141, pp. 245–259. Springer, Berlin (1985) CrossRefGoogle Scholar
  18. 18.
    Martin-Löf, P.: The definition of random sequences. Inf. Control 9(6), 602–619 (1966) CrossRefMATHGoogle Scholar
  19. 19.
    Merkle, W., Mihailović, N.: On the construction of effective random sets. In: Diks, K., Rytter, W. (eds.) Proceedings of Mathematical Foundations of Computer Science 2002, 27th International Symposium, MFCS 2002, Warsaw, Poland, August 26–30, 2002. Lecture Notes in Computer Science, vol. 2420, pp. 568–580. Springer, Berlin (2002). ISBN 3-540-44040-2 Google Scholar
  20. 20.
    Miyabe, K.: Truth-table Schnorr randomness and truth-table reducible randomness. Math. Log. Q. 57(3), 323–338 (2011) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Miyabe, K., Rute, J.: Van Lambalgen’s Theorem for uniformly relative Schnorr and computable randomness. In: Downey, R., Brendle, J., Goldblatt, R., Kim, B. (eds.) Proceedings of the 12th Asian Logic Conference, pp. 251–270 (2013) CrossRefGoogle Scholar
  22. 22.
    Nies, A.: Lowness properties and randomness. Adv. Math. 197, 274–305 (2005) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Nies, A.: Computability and Randomness. Oxford University Press, London (2009) CrossRefMATHGoogle Scholar
  24. 24.
    Schnorr, C.P.: Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, vol. 218. Springer, Berlin (1971) CrossRefMATHGoogle Scholar
  25. 25.
    Terwijn, S.A., Zambella, D.: Computational randomness and lowness. J. Symb. Log. 66(3), 1199–1205 (2001) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Weihrauch, K.: Computable Analysis: An Introduction. Springer, Berlin (2000) CrossRefGoogle Scholar
  27. 27.
    Weihrauch, K., Grubba, T.: Elementary computable topology. J. Univers. Comput. Sci. 15(6), 1381–1422 (2009) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations