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On the construction of effectively random sets

Published online by Cambridge University Press:  12 March 2014

Wolfgang Merkle
Affiliation:
Institut für Informatik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, D–69120 Heidelberg, Germany, E-mail: merkle@math.uni-heidelberg.de
Nenad Mihailović
Affiliation:
Institut für Informatik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, D–69120 Heidelberg, Germany, E-mail: mihailovic@math.uni-heidelberg.de

Abstract.

We present a comparatively simple way to construct Martin-Löf random and rec-random sets with certain additional properties, which works by diagonalizing against appropriate martingales. Reviewing the result of Gács and Kučera, for any given set X we construct a Martin-Löf random set from which X can be decoded effectively.

By a variant of the basic construction we obtain a rec-random set that is weak truth-table autoreducible and we observe that there are Martin-Löf random sets that are computably enumerable self-reducible. The two latter results complement the known facts that no rec-random set is truth-table autoreducible and that no Martin-Löf random set is Turing-autoreducible [8, 24].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

[1]Ambos-Spies, K. and Kučera, A., Randomness in computability theory, Computability Theory and its Applications – Current Trends and Open Problems (Cholak, P. A., Lempp, S., Lerman, M., and Shore, R. A., editors), Contemporary Mathematics, vol. 257, American Mathematical Society, 2000, pp. 114.CrossRefGoogle Scholar
[2]Ambos-Spies, K. and Mayordomo, E., Resource-bounded measure and randomness, Complexity, Logic, and Recursion Theory (Sorbi, A., editor), Dekker, New York, 1997, pp. 147.Google Scholar
[3]Apostol, T. M., Mathematical Analysis, 3rd ed., Addison Wesley, 1978.Google Scholar
[4]Balcázar, J. L., Díaz, J., and Gabarró, J., Structural Complexity I, Springer, 1995.CrossRefGoogle Scholar
[5]Calude, C. S., A characterization of c.e. random reals, Theoretical Computer Science, vol. 271 (2002), pp. –14.CrossRefGoogle Scholar
[6]Calude, C. S., Information and Randomness, 2nd ed., Springer-Verlag, 2002.CrossRefGoogle Scholar
[7]Ebert, T., Applications of recursive operators to randomness and complexity, Ph.D. thesis, University of California at Santa Barbara, 1998.Google Scholar
[8]Ebert, T., Merkle, W., and Vollmer, H., On the autoreducibility of random sequences, SIAM Journal on Computing, vol. 32 (2003), pp. 15421569.CrossRefGoogle Scholar
[9]Gács, P., Every sequence is reducible to a random one, Information and Control, vol. 70 (1986), pp. 186192.CrossRefGoogle Scholar
[10]Hertling, P., Surjective functions on computably growing Cantor sets, Journal of Universal Computer Science, vol. 3 (1996), pp. 12261240.Google Scholar
[11]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, 1985, pp. 245259.CrossRefGoogle Scholar
[12]Kučera, A., On the use of diagonally nonrecursive functions, Logic Colloquium'87 (Ebbinghaus, H. D.et al., editors), Studies in Logic and the Foundations of Mathematics, vol. 129, North-Holland, 1989, pp. 219239.Google Scholar
[13]Li, M. and Vitányi, P., An Introduction to Kolmogorov Complexity and its Applications, 2nd ed., Springer, 1997.CrossRefGoogle Scholar
[14]Lutz, J. H., Almost everywhere high nonuniform complexity, Journal of Computer and System Sciences, vol. 44 (1992), pp. 220258.CrossRefGoogle Scholar
[15]Lutz, J. H., The quantitative structure of exponential time, Complexity Theory Retrospective II (Hemaspaandra, L. A. and Selman, A. L., editors), Springer, 1997, pp. 225260.CrossRefGoogle Scholar
[16]Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), no. 6, pp. 602619.CrossRefGoogle Scholar
[17]Mayordomo, E., Contributions to the study of resource-hounded measure, Ph.D. thesis, Universitat Politècnica de Catalunya, Barcelona, Spain, 1994.Google Scholar
[18]Merkle, W., The complexity of stochastic sequences, Proceedings of the IEEE Conference on Computational Complexity, IEEE Computer Society Press, 2003, pp. 230235.Google Scholar
[19]Odifreddi, P., Classical Recursion Theory, North-Holland, Amsterdam, 1989.Google Scholar
[20]Schnorr, C.-P., A unified approach to the definition of random sequences, Mathematical Systems Theory, vol. 5 (1971), pp. 246258.CrossRefGoogle Scholar
[21]Schnorr, C.-P., Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics, vol. 218, Springer, 1971, in German.CrossRefGoogle Scholar
[22]Soare, R. I., Recursively Enumerable Sets and Degrees, Springer, 1987.CrossRefGoogle Scholar
[23]Terwijn, S. A., Computability and measure, Ph.D. thesis, Universiteit van Amsterdam, Amsterdam, Netherlands, 1998.Google Scholar
[24]Trakhtenbrot, B. A., On autoreducibility, Soviet Mathematics Doklady, vol. 11 (1970), pp. 814817.Google Scholar
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