Van Lambalgen's Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen's Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that is high. We extend this result to that is . We also prove that there exists A such that, for each n, the real is for some universal Turing machine M by using the extended van Lambalgen's Theorem.
References
[1] Buhrman, H., and L. Longpré, "Compressibility and resource bounded measure", pp. 13--24 in STACS 96 (Grenoble, 1996), vol. 1046 of Lecture Notes in Computer Science, Springer, Berlin, 1996. MR1462082 1015.68082[1] Buhrman, H., and L. Longpré, "Compressibility and resource bounded measure", pp. 13--24 in STACS 96 (Grenoble, 1996), vol. 1046 of Lecture Notes in Computer Science, Springer, Berlin, 1996. MR1462082 1015.68082
[2] Buhrman, H., D. van Melkebeek, K. W. Regan, D. Sivakumar, and M. Strauss, "A generalization of resource-bounded measure, with application to the BPP" vs. EXP problem, SIAM Journal on Computing, vol. 30 (2000), pp. 576--601. MR1769372 0963.68230 10.1137/S0097539798343891[2] Buhrman, H., D. van Melkebeek, K. W. Regan, D. Sivakumar, and M. Strauss, "A generalization of resource-bounded measure, with application to the BPP" vs. EXP problem, SIAM Journal on Computing, vol. 30 (2000), pp. 576--601. MR1769372 0963.68230 10.1137/S0097539798343891
[4] Downey, R., D. R. Hirschfeldt, J. S. Miller, and A. Nies, "Relativizing Chaitin's halting probability", Journal of Mathematical Logic, vol. 5 (2005), pp. 167--92. MR2188515 1093.03025 10.1142/S0219061305000468[4] Downey, R., D. R. Hirschfeldt, J. S. Miller, and A. Nies, "Relativizing Chaitin's halting probability", Journal of Mathematical Logic, vol. 5 (2005), pp. 167--92. MR2188515 1093.03025 10.1142/S0219061305000468
[5] Kučera, A., "Measure, $\Pi^0_1$"-classes and complete extensions of $\rm PA$, pp. 245--59 in Recursion Theory Week (Oberwolfach, 1984), vol. 1141 of Lecture Notes in Mathematics, Springer, Berlin, 1985. MR820784 0622.03031[5] Kučera, A., "Measure, $\Pi^0_1$"-classes and complete extensions of $\rm PA$, pp. 245--59 in Recursion Theory Week (Oberwolfach, 1984), vol. 1141 of Lecture Notes in Mathematics, Springer, Berlin, 1985. MR820784 0622.03031
[6] van Lambalgen, M., Random Sequences, Ph.D. thesis, University of Amsterdam, Amsterdam, 1987. 0628.60001[6] van Lambalgen, M., Random Sequences, Ph.D. thesis, University of Amsterdam, Amsterdam, 1987. 0628.60001
[7] Li, M., and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, 3d edition, Graduate Texts in Computer Science. Springer, New York, 2008. Also second edition (1997). MR2494387 MR1438307 1185.68369 0866.68051[7] Li, M., and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, 3d edition, Graduate Texts in Computer Science. Springer, New York, 2008. Also second edition (1997). MR2494387 MR1438307 1185.68369 0866.68051
[9] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, Oxford, 2009. MR2548883 1169.03034[9] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, Oxford, 2009. MR2548883 1169.03034
[10] Odifreddi, P., Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers, vol. 125 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1989. MR982269 0661.03029[10] Odifreddi, P., Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers, vol. 125 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1989. MR982269 0661.03029
[11] Odifreddi, P. G., Classical Recursion Theory. Vol. II, vol. 143 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1999. MR1718169 0931.03057[11] Odifreddi, P. G., Classical Recursion Theory. Vol. II, vol. 143 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1999. MR1718169 0931.03057
[12] Schnorr, C.-P., "A unified approach to the definition of random sequences", Mathematical Systems Theory, vol. 5 (1971), pp. 246--58. MR0354328 0227.62005 10.1007/BF01694181[12] Schnorr, C.-P., "A unified approach to the definition of random sequences", Mathematical Systems Theory, vol. 5 (1971), pp. 246--58. MR0354328 0227.62005 10.1007/BF01694181
[13] Soare, R. I., Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. MR882921 0623.03042 0667.03030[13] Soare, R. I., Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. MR882921 0623.03042 0667.03030
[14] Ville, J., Étude Critique de la Notion du Collectif, vol. 3 of Monographies des Probabilités, Gauthier-Villars, Paris, 1939. 0021.14601 0021.14505[14] Ville, J., Étude Critique de la Notion du Collectif, vol. 3 of Monographies des Probabilités, Gauthier-Villars, Paris, 1939. 0021.14601 0021.14505
