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Schnorr randomness

  • Rodney G. Downey (a1) and Evan J. Griffiths (a2)

Abstract.

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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[1]Ambos-Spies, K. and Kučera, A., Randomness in computability theory, Computability theory and its applications (Cholak, Lempp, , Lerman, , and Shore, , editors). Contemporary Mathematics, vol. 257, American Mathematical Society, 2000, pp. 114.
[2]Ambos-Spies, K. and Mayordomo, E., Resource bounded measure and randomness, Complexity, logic and recursion theory (Sorbi, A., editor), Marcel-Decker, New York, 1997, pp. 148.
[3]Chaitin, G., A theory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329340.
[4]Downey, R., Griffiths, E., and LaForte, G., On Schnorr and computable randomness, martingales, and machines, Mathematical Logic Quarterly, to appear.
[5]Downey, R., Hirschfeldt, D., Nies, A., and Stephan, F., 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. 3755, 07, 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.
[6]Kolmogorov, A. N., Three approaches to the quantitative definition of information, Problems of Information Transmission (Problemy Peredachi Informatsii), vol. 1 (1965), pp. 17.
[7]Kučera, A., Measure, -classes and complete extensions of PA, Recursion theory week (Ebbinghaus, H.-D., Müller, G. H., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 245259.
[8]Kučera, A. and Slaman, T., Randomness and recursive enumerability, SI AM Journal on Computing, vol. 31 (2001), pp. 199211.
[9]van Lambalgen, M., Random sequences, Ph.D. thesis, University of Amsterdam, 1987.
[10]Levin, L., Measures of complexity of finite objects (axiomatic description), Soviet Mathematics Doklady, vol. 17 (1976), pp. 522526.
[11]Li, M. and Vitanyi, P., An introduction to Kolmogorov complexity and its applications, 2nd ed., Springer-Verlag, New York, 1997.
[12]Lutz, J. H., Almost everywhere high nonuniform complexity, Journal of Computer and System Sciences, vol. 44 (1992), pp. 220258.
[13]Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602619.
[14]Nies, A., Lowness properties and randomness, to appear.
[15]Schnorr, C. P., Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics, vol. 218, Springer-Verlag, Berlin, New York, 1971.
[16]Schnorr, C. P., Process complexity and effective random tests, Journal of Computer and System Sciences, vol. 7 (1973), pp. 376388.
[17]Solomonof, R., A formal theory of inductive inference, Part I, Information and Control, vol. 7 (1964), pp. 122.
[18]Solovay, R., Draft of paper (or series of papers) on Chaitin's work. Unpublished, 215 pages, 05 1975.
[19]Terwijn, A. and Zambella, D., Computational randomness and lowness, this Journal, vol. 66 (2001), pp. 11991205.
[20]Wang, Y., Randomness and complexity, Ph.D. thesis, University of Heidelberg, 1996.

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Schnorr randomness

  • Rodney G. Downey (a1) and Evan J. Griffiths (a2)

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