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The Kolmogorov-Loveland stochastic sequences are not closed under selecting subsequences

Published online by Cambridge University Press:  12 March 2014

Wolfgang Merkle
Wolfgang Merkle*
Affiliation:
Ruprecht-Karls-Universität Heidelberg, Institut Für Informatik, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany, E-mail: [email protected]
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Abstract

It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the question whether the Kolmogorov-Loveland stochastic sequences are closed under selecting sequences by Kolmogorov-Loveland selection rules, i.e., by not necessarily monotonic, partial computable selection rules. The following previously known results are obtained as corollaries. The Mises-Wald-Church stochastic sequences are not closed under computable permutations, hence in particular they form a strict superclass of the class of Kolmogorov-Loveland stochastic sequences. The Kolmogorov-Loveland selection rules are not closed under composition.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1362 - 1376
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Ambos-Spies, K., Algorithmic randomness revisited, Language, Logic and Formalization of Knowledge, Coimbra Lecture and Proceedings of a Symposium held in Siena in September 1997 (McGuinness, B., editor), Bibliotheca, 1998, pp. 33–52.Google Scholar
[2] Ambos-Spies, K. and Kučera, A., Randomness in computability theory, Computability theory: Current Trends and Open Problems (Cholak, P. et al., editors), Contemporary Mathematics, vol. 257, American Mathematical Society, 2000, pp. 1–14.Google Scholar
[3] Ambos-Spies, K. and Mayordomo, E., Resource-bounded measure and randomness, Complexity, logic, and recursion theory (Sorbi, A., editor), Lecture Notes in Pure and Applied Mathematics, vol. 187, Marcel Dekker Inc., New York, 1997, pp. 1–47.Google Scholar
[4] Ambos-Spies, K., Mayordomo, E., Wang, Y., and Zheng, X., Resource-bounded balanced genericity, stochasticity and weak randomness, 13th Annual Symposium on Theoretical Aspects of Computer Science, STACS'96 (Puech, C. and Reischuk, R., editors), Lecture Notes in Computer Science, vol. 1046, Springer, 1997, pp. 63–74.Google Scholar
[5] Balcázar, J. L., Díaz, J., and Gabarró, J., Structural complexity. Vol. I and II, Springer-Verlag, 1995 and 1990.Google Scholar
[6] Church, A., On the concept of a random sequence, Bulletin of the American Mathematical Society, vol. 46 (1940), pp. 130–135.Google Scholar
[7] Daley, R. P., Minimal-program complexity of pseudo-recursive and pseudo-random sequences, Mathematical Systems Theory, vol. 9 (1975), pp. 83–94.Google Scholar
[8] Feller, W., An introduction to probability theory and its applications. Vol. I, third ed., revised printing, John Wiley & Sons Inc., 1968.Google Scholar
[9] Kolmogorov, A. N., On tables of random numbers, Sankhyā, Ser. A, vol. 25 (1963), pp. 369–376, Reprinted in Theoretical Computer Science, vol. 207 (1998), pp. 387–395.Google Scholar
[10] Kolmogorov, A. N. and Uspensky, V. A., Algorithms and randomness, Theory of Probability and its Applications, vol. 32 (1987), pp. 389–412.Google Scholar
[11] van Lambalgen, M., Random sequences, Ph.D. thesis , University of Amsterdam, Amsterdam, 1987.Google Scholar
[12] Li, M. and Vitányi, P., An Introduction to Kolmogorov Complexity and Its Applications, second ed., Springer, 1997.Google Scholar
[13] Loveland, D. W., The Kleene hierarchy classification of recursively random sequences, Transactions of the American Mathematical Society, vol. 125 (1966), pp. 497–510.Google Scholar
[14] Loveland, D. W., A new interpretation of the von Mises' concept of random sequence, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 279–294.CrossRefGoogle Scholar
[15] Lutz, J. H., Private communication, 03 2002.Google Scholar
[16] Lutz, J. H. and Schweizer, D. L., Feasible reductions to Kolmogorov-Loveland stochastic sequences, Theoretical Computer Science, vol. 225 (1999), pp. 185–194.Google Scholar
[17] Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.Google Scholar
[18] Merkle, W., The complexity of stochastic sequences, Conference on computational complexity, IEEE Computer Society Press, 2003, pp. 230–235.Google Scholar
[19] Muchnik, An. A., Semenov, A. L., and Uspensky, V. A., Mathematical metaphysics of randomness, Theoretical Computer Science, vol. 207 (1998), pp. 263–317.Google Scholar
[20] Odifreddi, P., Classical recursion theory. Vol. I, North-Holland Publishing Company, Amsterdam, 1989.Google Scholar
[21] Papadimitriou, C. H., Computational complexity, Addison-Wesley Publishing Company, 1994.Google Scholar
[22] Schnorr, C. P., Zufälligkeit und Wahrscheinlichkeit, in German, Lecture Notes in Mathematics, vol. 218, Springer-Verlag, 1971.Google Scholar
[23] Shen', A. Kh., The frequency approach to the definition of the notion of a random sequence (in Russian), Semiotika iInformatika, vol. 18 (1982), pp. 14–41.Google Scholar
[24] Shen', A. Kh., On relations between different algorithmic definitions of randomness, Soviet Mathematics Doklady, vol. 38 (1988), pp. 316–319.Google Scholar
[25] Shen', A. Kh., Private communication, 11 2001.Google Scholar
[26] Shen', A. Kh., Private communication, 03 2002.Google Scholar
[27] Uspensky, V. A., Semenov, A. L., and Shen', A. Kh., Can an individual sequence of zeros and ones be random?, Russian Mathematical Surveys, vol. 45 (1990), pp. 121–189.Google Scholar
[28] Ville, J., Étude critique de la notion de collectif, in French, Gauthiers-Villars, Paris, 1939.Google Scholar