Journal of Symbolic Logic

Schnorr randomness

Rodney G. Downey and Evan J. Griffiths

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Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.

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J. Symbolic Logic, Volume 69, Issue 2 (2004), 533-554.

First available in Project Euclid: 19 April 2004

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Downey, Rodney G.; Griffiths, Evan J. Schnorr randomness. J. Symbolic Logic 69 (2004), no. 2, 533--554. doi:10.2178/jsl/1082418542.

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  • K. Ambos-Spies and A. Kučera Randomness in computability theory, Computability theory and its applications (Cholak, Lempp, Lerman, and Shore, editors), Contemporary Mathematics, vol. 257, American Mathematical Society,2000, pp. 1--14.
  • K. Ambos-Spies and E. Mayordomo Resource bounded measure and randomness, Complexity, logic and recursion theory (A. Sorbi, editor), Marcel-Decker, New York,1997, pp. 1--48.
  • G. Chaitin A theory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329--340.
  • R. Downey, E. Griffiths, and G. LaForte On Schnorr and computable randomness, martingales, and machines, Mathematical Logic Quarterly, to appear.
  • R. Downey, D. Hirschfeldt, A. Nies, and F. Stephan Trivial reals, extended abstract in Computability and Complexity in Analysis, Malaga, Electronic Notes in Theoretical Computer Science, and proceedings, edited by Brattka, Schröder, Weihrauch, FernUniversität Hagen, 294-6/2002, pp. 37-55, July, 2002. Final version appears in Proceedings of the 7th and 8th Asian Logic Conferences (Rod Downey, Ding Decheng, Tung Shi Ping, Qiu Yu Hui, Mariko Yasugi, and Wu Guohua, editors) World Scientific, 2003, pp. 103-131.
  • A. N. Kolmogorov Three approaches to the quantitative definition of information, Problems of Information Transmission (Problemy Peredachi Informatsii), vol. 1 (1965), pp. 1--7.
  • A. Kučera Measure, $\Pi^0_1$-classes and complete extensions of $PA$, Recursion theory week (H.-D. Ebbinghaus, G. H. Müller, and G. E. Sacks, editors), Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, Berlin, Heidelberg, New York,1985, pp. 245--259.
  • A. Kučera and T. Slaman Randomness and recursive enumerability, SIAM Journal on Computing, vol. 31 (2001), pp. 199--211.
  • M. van Lambalgen Random sequences, Ph.D. thesis, University of Amsterdam,1987.
  • L. Levin Measures of complexity of finite objects (axiomatic description), Soviet Mathematics Doklady, vol. 17 (1976), pp. 522--526.
  • M. Li and P. Vitanyi An introduction to Kolmogorov complexity and its applications, 2nd ed., Springer-Verlag, New York,1997.
  • J. H. Lutz Almost everywhere high nonuniform complexity, Journal of Computer and System Sciences, vol. 44 (1992), pp. 220--258.
  • P. Martin-Löf The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602--619.
  • A. Nies Lowness properties and randomness, to appear.
  • C. P. Schnorr Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics, vol. 218, Springer-Verlag, Berlin, New York,1971.
  • R. Solomonof A formal theory of inductive inference, Part I, Information and Control, vol. 7 (1964), pp. 1--22.
  • R. Solovay Draft of paper (or series of papers) on Chaitin's work, Unpublished, 215 pages, May 1975.
  • A. Terwijn and D. Zambella Computational randomness and lowness, Journal of Symbolic Logic, vol. 66 (2001), pp. 1199--1205.
  • Y. Wang Randomness and complexity, Ph.D. thesis, University of Heidelberg,1996.