Abstract
We investigate the characterizations of effective randomness in terms of Martin-Löf covers and martingales. First, we address a question of Ambos-Spies and Kučera [1], who asked for a characterization of computable randomness in terms of tests. We argue that computable randomness can be characterized in term of Martin-Löf tests and effective probability distributions on Cantor space.
Second, we show that the class of Martin-Löf random sets coincides with the class of sets of reals that are random with respect to computable martingale processes. This improves on results of Hitchcock and Lutz [8], who showed that the latter class is contained in the class of Martin-Löf random sets and is a strict superset of the class of rec-random sets.
Third, we analyze the sequence of measures of the components of a universal Martin-Löf test. Kučera and Slaman [12] showed that any component of a universal Martin-Löf test defines a class of Martin-Löf random measure. Further, since the sets in a Martin-Löf test are uniformly computably enumerable, so is the corresponding sequence of measures. We prove an exact converse and hence a characterization. For any uniformly computably enumerable sequence r 1,r 2,... of reals such that each r i is Martin-Löf random and less than 2 − − i there is a universal Martin-Löf test U 1, U 2,... such that U i { 0,1} ∞ has measure r i .
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ambos-Spies, K., Kučera, A.: Randomness in computability theory. In: Cholak, P.A., et al. (eds.) Computability Theory and Its Applications. Current Trends and Open Problems. Contemporary Mathematics, vol. 257, pp. 1–14. American Mathematical Society (AMS), Providence (2000)
Ambos-Spies, K., Mayordomo, E.: Resource-bounded measure and randomness. In: Sorbi, A. (ed.) Complexity, Logic, and Recursion Theory, pp. 1–47. Dekker, New York (1997)
Balcázar, J.L., D´ıaz, J., Gabarró, J.: Structural Complexity I. Springer, Heidelberg (1995)
Calude, C.S.: Information and Randomness. Springer, Heidelberg (1994)
Calude, C.S.: A characterization of c.e. random reals. Theoretical Computer Science 271, 3–14 (2002)
Durrett, R.: Essentials of Stochastic Process. Springer, Heidelberg (1999)
Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity (2003) (manuscript)
Hitchcock, J.M., Lutz, J.H.: Why computational complexity requires stricter martingales. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 549–560. Springer, Heidelberg (2002)
Hitchcock, J.M.: Small Spans in Scaled Dimension. In: IEEE Conference on Computational Complexity (2004) (to appear)
Kučera, A.: Measure, \(\rm \Pi^{0}_{1}\) -classes and complete extensions of PA. In: Ebeling, W., Rechenberg, I., Voigt, H.-M., Schwefel, H.-P. (eds.) PPSN 1996. LNCS, vol. 1141, pp. 245–259. Springer, Heidelberg (1996)
Kučera, A.: On the use of diagonally nonrecursive functions. In: Ebbinghaus, H.-D., et al. (eds.) Logic Colloquium 1987. Studies in Logic and the Foundations of Mathematics, vol. 129, pp. 219–239. North-Holland, Amsterdam (1989)
Kučera, A., Slaman, T.A.: Randomness and recursive enumerability. SIAM Journal on Computing 31(1), 199–211 (2001)
Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 2nd edn. Springer, Heidelberg (1997)
Lutz, J.H.: Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences 44, 220–258 (1992)
Lutz, J.H.: The quantitative structure of exponential time. In: Hemaspaandra, L.A., Selman, A.L. (eds.) Complexity Theory Retrospective II, pp. 225–260. Springer, Heidelberg (1997)
Martin-Löf, P.: The definition of random sequences. Information and Control 9(6), 602–619 (1966)
Mayordomo, E.: Contributions to the Study of Resource-Bounded Measure. Doctoral dissertation, Universitat Politècnica de Catalunya, Barcelona, Spain (1994)
Muchnik, A.A., Semenov, A.L., Uspensky, V.A.: Mathematical metaphysics of randomness. Theoretical Computer Science 207, 263–317 (1990)
Odifreddi, P.: Classical Recursion Theory. North-Holland, Amsterdam (1989)
Schnorr, C.-P.: A unified approach to the definition of random sequences. Mathematical Systems Theory 5, 246–258 (1971)
Schnorr, C.-P.: Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, vol. 218. Springer, Heidelberg (1971)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)
Solovay, R.M.: Draft of a Paper (or Series of Papers) on Chaitin’s Work...Manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY (1974)
Terwijn, S.A.: Computability and Measure. Doctoral dissertation, Universiteit van Amsterdam, Amsterdam, Netherlands (1998)
Wang, Y.: A Separation of two randomness concepts. Information Processing Letters 69, 115–118 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Merkle, W., Mihailović, N., Slaman, T.A. (2004). Some Results on Effective Randomness. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_82
Download citation
DOI: https://doi.org/10.1007/978-3-540-27836-8_82
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22849-3
Online ISBN: 978-3-540-27836-8
eBook Packages: Springer Book Archive