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ω-CHANGE RANDOMNESS AND WEAK DEMUTH RANDOMNESS

  • JOHANNA N. Y. FRANKLIN (a1) and KENG MENG NG (a2)

Abstract

We extend our work on difference randomness. Each component of a difference test is a Boolean combination of two r.e. open sets; here we consider tests in which the kth component is a Boolean combination of g(k) r.e. open sets for a given recursive function g. We use this method to produce an alternate characterization of weak Demuth randomness in terms of these tests and further show that a real is weakly Demuth random if and only if it is Martin-Löf random and cannot compute a strongly prompt r.e. set. We conclude with a study of related lowness notions and obtain as a corollary that lowness for balanced randomness is equivalent to being recursive.

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[1]Ambos-Spies, K., Jockusch, C., Shore, R., and Soare, R.. An algebraic decomposition of recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Transactions of the American Mathematical Society, vol. 281 (1984), pp. 109128.
[2]Barmpalias, George, Miller, Joseph S., and Nies, André. Randomness notions and partial relativization. Israel Journal of Mathematics, vol. 191 (2012), no. 2, pp. 791816.
[3]Bienvenu, Laurent, Day, Adam R., Greenberg, Noam, Kučera, Antonín, Miller, Joseph S., Nies, André, and Turetsky, Dan. Computing K-trivial sets by incomplete random sets. Bulletin of Symbolic Logic. To appear.
[4]Demuth, Osvald. On some classes of arithmetical real numbers. Commentationes Mathematicae Universitatis Carolinae, vol. 23 (1982), pp. 453465. In Russian.
[5]Diamondstone, David and Ng, Keng Meng. Strengthening prompt simplicity, this Journal, vol. 76 (2011), no, 3, pp. 946–972.
[6]Downey, Rod, Nies, André, Weber, Rebecca, and Yu, Liang. Lowness and ${\rm{\Pi }}_2^0 $nullsets, this Journal, vol. 71 (2006), no. 3, pp. 10441052.
[7]Downey, Rodney G. and Hirschfeldt, Denis R.. Algorithmic randomness and complexity, Springer, New York, 2010.
[8]Figueira, Santiago, Hirschfeldt, Denis, Miller, Joseph S., Ng, Keng Meng, and Nies, André. Counting the changes of random${\rm{\Delta }}_2^0 $sets, Programs, proofs, processes, Lecture Notes in Computer Science, vol. 6158, Springer, Berlin, 2010, pp. 162171.
[9]Franklin, Johanna N. Y. and Ng, Keng Meng. Difference randomness. Proceedings of the American Mathematical Society, vol. 139 (2011), no. 1, pp. 345360.
[10]Greenberg, Noam and Nies, André. Benign cost functions and lowness properties, this Journal, vol. 76 (2011), no. 1, pp. 289–312.
[11]Hirschfeldt, D. and Miller, J.. Unpublished.
[12]Kučera, Antonín and Nies, André. Demuth randomness and computational complexity. Annals of Pure and Applied Logic, vol. 162 (2011), no. 7, pp. 504513.
[13]Nies, André. Computability and randomness, Clarendon Press, Oxford, 2009.
[14]Nies, André, Stephan, Frank, and Terwijn, Sebastiaan A.. Randomness, relativization and Turing degrees, this Journal, vol. 70 (2005), no. 2, pp. 515–535.
[15]Soare, Robert I.. Recursively enumerable sets and degrees, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987.
[16]Stephan, Frank. Martin-Löf random and PA-complete sets, Technical Report 58, Matematisches Institut, Universität Heidelberg, Heidelberg, 2002.

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