Abstract
A Schnorr test relative to some oracle A may informally be called “universal” if it covers all Schnorr tests. Since no true universal Schnorr test exists, such an A cannot be computable. We prove that the sets with this property are exactly those with high Turing degree. Our method is closely related to the proof of Terwijn and Zambella’s characterization of the oracles which are low for Schnorr tests. We also consider the oracles which compute relativized Schnorr tests with the weaker property of covering all computable reals. The degrees of these oracles strictly include the hyperimmune degrees and are strictly included in the degrees not computably traceable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bartoszyński T.: Additivity of measure implies additivity of category. Trans. Am. Math. Soc. 281, 225–239 (1987)
Bartoszyński T., Judah H.: Set Theory: On the Structure of the Real Line. A K Peters, Wellesley (1995)
Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. I, Chap. 5. Springer, Berlin (to appear)
Downey R.G., Griffiths E.J.: Schnorr randomness. J. Symb. Log. 69, 533–554 (2004)
Downey R.G., Merkle W., Reimann J.: Schnorr dimension. Math. Struct. Comp. Sci. 16(5), 789–811 (2006)
Goldstern M., Judah H., Shelah S.: Strong measure zero sets without Cohen reals. J. Symb. Log. 58, 1323–1341 (1993)
Jockusch, C.G. Jr., Soare, R.I.: \({\Pi_1^0}\) classes and degrees of theories. Trans. Am. Math. Soc. 33–56 (1972)
Kautz, S.A.: Degrees of random sets. Ph.D thesis, Cornell University (1991)
Kunen K.: Random and cohen reals. In: Kunen, K., Vaughan, J.E. (eds) Handbook of Set-Theoretic Topology, Chapter 20, pp. 887–911. Elsevier Science, Amsterdam (1983)
Kurtz, S.: Randomness and genericity in the degrees of unsolvability. Ph.D thesis, University of Illinois, Urbana (1981)
Martin D.A.: Classes of recursively enumerable sets and degrees of unsolvability. Z. Math. Log. Grund. Math. 12, 295–310 (1966)
Martin-Löf P.: The definition of random sequences. Inf. Control 9, 602–619 (1966)
Miller A.: Additivity of measure implies dominating reals. Proc. Am. Math. Soc. 91, 111–117 (1984)
Nies A., Stephan F., Terwijn S.A.: Randomness, relativization, and the turing degrees. J. Symb. Log. 70, 515–535 (2005)
Pawlikowski J., Recław I.: Parametrized Cichoń’s diagram. Fund. Math. 147, 135–155 (1995)
Raisonnier J.: A mathematical proof of S. Shelah’s theorem on the measure problem and related results. Isr. J. Math. 48, 48–56 (1984)
Rothberger F.: Eine Äquivalenz zwischen der Kontinuumhypothese und der Existenz der Lusinschen und Sierpińskischen Mengen. Fund. Math. 30, 215–217 (1938)
Schnorr C.-P.: Zufälligkeit und Wahrscheinlichkeit: Lecture Notes in Mathematics, vol. 218. Springer, Berlin (1971)
Schnorr C.-P.: Process complexity and effective random tests. J. Comp. Syst. Sci. 7, 376–388 (1973)
Stephan, F., Yu, L.: Lowness for weakly 1-generic and Kurtz random. In: Theory and Applications of Models of Computation, vol. 3959, pp. 756–764. Springer, Berlin (2006)
Terwijn S.A., Zambella D.: Computational randomness and lowness. J. Symb. Log. 66, 1199–1205 (2001)
van Lambalgen M.: Von Mises’ notion of random sequence reconsidered. J. Symb. Log. 52, 725–755 (1987)
Yu L.: Lowness for genericity. Arch. Math. Log. 45, 233–238 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rupprecht, N. Relativized Schnorr tests with universal behavior. Arch. Math. Logic 49, 555–570 (2010). https://doi.org/10.1007/s00153-010-0187-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-010-0187-6