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Jump inversions inside effectively closed sets and applications to randomness

  • George Barmpalias (a1), Rod Downey (a2) and Keng Meng Ng (a3)


We study inversions of the jump operator on classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees above 0′ which are not 0′-dominated. A further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails.

Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable.



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[1]Barmpalias, George, Miller, Joseph S., and Nies, André, Randomness notions and partial relativization, submitted.
[2]Cenzer, D. and Remmel, J. B., classes in mathematics, Handbook of recursive mathematics – Volume 2, Recursive algebra, analysis and combinatorics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., Remmel, J. B., and Marek, V. W., editors), Studies in Logic and the Foundations of Mathematics, 139, Elsevier, 1998, pp. 623821.
[3]Cenzer, Douglas, classes in computability theory, Handbook of computability theory (Griffor, E. R., editor), Studies in Logic and the Foundations of Mathematics, vol. 140, North-Holland, Amsterdam, 1999, pp. 3785.
[4]Cooper, S. B., Minimal degrees and the jump operator, this Journal, vol. 38 (1973), pp. 249271.
[5]Downey, Rod and Greenberg, Noam, Turing degrees of reals of positive packing dimension, Information Processing Letters, vol. 108 (2008), pp. 198203.
[6]Downey, Rod and Hirschfeldt, Denis, Algorithmic randomness and complexity, Theory and Applications of Computability, Springer-Verlag, 2010.
[7]Downey, Rod, Jockusch, Carl G. Jr., and Stob, Michael, Array nonrecursive sets andgenericity, Computability, enumerability, unsolvability: Directions in recursion theory, London Mathematical Society Lecture Notes Series, vol. 224, Cambridge University Press, 1996, pp. 93104.
[8]Downey, Rod and Miller, Joseph S., A basis theorem for classes of positive measure and jump inversion for random reals, Proceedings of the American Mathematical Society, vol. 134 (2006), no. 1, pp. 283288, (electronic).
[9]Downey, Rod, Nies, André, Weber, Rebecca, and Yu, Liang, Lowness and null sets, this Journal, vol. 71 (2006), pp. 10441052.
[10]Friedberg, R. M., A criterion for completeness of degrees of unsolvability, this Journal, vol. 22 (1957), pp. 159160.
[11]Gács, Péter, Every sequence is reducible to a random one, Information and Control, vol. 70 (1986), no. 2–3, pp. 186192.
[12]Gaifman, Haim and Snir, Marc, Probabilities over rich languages, testing and randomness, this Journal, vol. 47 (1982), no. 3, pp. 495548.
[13]Jockusch, C. Jr. and Stephan, Frank, A cohesive set which is not high, Mathematical Logic Quarterly, vol. 39 (1993), pp. 515530.
[14]Jockusch, C. Jr. and Stephan, Frank, Correction to “a cohesive set which is not high”, Mathematical Logic Quarterly, vol. 43 (1997), p. 569.
[15]Jockusch, Carl G. Jr. and Soare, Robert I., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.
[16]Kautz, S., Degrees of random sets, Ph.D. Dissertation, Cornell University, 1991.
[17]Kučera, Antonín, Measure, -classes and complete extensions of PA, Recursion theory week (Oberwolfach, 1984) (Ambos-Spies, K., Sacks, G. E., and Müller, G. H., editors), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245259.
[18]Kučera, Antonín, An alternative, priority-free, solution to Post's problem, Mathematical foundations of computer science, 1986 (Bratislava, 1986) (Gruska, J. and Wiedermann, J.Rovan, B., editors), Lecture Notes in Computer Science, vol. 233, Springer, Berlin, 1986, pp. 493500.
[19]Kurtz, S., Randomness and genericity in the degrees of unsolvability, Ph.D. Dissertation, University of Illinois, Urbana, 1981.
[20]Martin, D., Measure, category, and degrees of unsolvability, unpublished manuscript, 1960s.
[21]Martin, D. A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295310.
[22]Merkle, W., Miller, J., Nies, A., Reimann, J., and Stephan, F., Kolmogorov–Loveland randomness and stochasticity, Annals of Pure and Applied Logic, vol. 138 (2006), pp. 183210.
[23]Miller, Joseph S. and Nies, André, Randomness and computability: open questions, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 390410.
[24]Nies, André, Computability and randomness, Oxford University Press, 2009.
[25]Nies, André, Stephan, Frank, and Terwijn, Sebastiaan A., Randomness, relativizalion and Turing degrees, this Journal, vol. 70 (2005), no. 2, pp. 515535.
[26]Posner, David, The upper semilattice of degrees below 0′ is complemented, this Journal, vol. 46 (1981), pp. 705713.
[27]Sacks, G. E., Degrees of unsolvability, Annals of Mathematical Studies, vol. 55, Princeton University Press, 1963.
[28]Shoenfield, J., On degrees of unsolvability, Annals of Mathematics. Second Series, vol. 69 (1959), pp. 644653.
[29]Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.
[30]Stephan, F. and Yu, L., Lowness for weakly 1-generic and Kurtz-random, Proceedings of Theory and Applications of Models of Computation, Lecture Notes in Computer Science, vol. 3959, Springer, 2006, pp. 756764.
[31]Stephan, Frank, Martin-Löf random and PK-complete sets, Logic colloquium '02 (Chatzidakis, Z., Koepke, P., and Pohlers, W., editors), Lecture Notes in Logic, vol. 27, Association for Symbolic Logic, La Jolla, CA, 2006, pp. 342348.
[32]Stillwell, John, Decidability of the “almost all” theory of degrees, this Journal, vol. 37 (1972), pp. 501506.
[33]Yu, Liang, When van Lambalgen's theorem fails, Proceedings of the American Mathematical Society, vol. 135 (2007), no. 3, pp. 861864, (electronic).

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Jump inversions inside effectively closed sets and applications to randomness

  • George Barmpalias (a1), Rod Downey (a2) and Keng Meng Ng (a3)


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