Journal of Symbolic Logic

Benign cost functions and lowness properties

Noam Greenberg and André Nies

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We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of Diamondstone's and Ng's regarding cupping with superlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees.

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J. Symbolic Logic, Volume 76, Issue 1 (2011), 289-312.

First available in Project Euclid: 4 January 2011

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Greenberg, Noam; Nies, André. Benign cost functions and lowness properties. J. Symbolic Logic 76 (2011), no. 1, 289--312. doi:10.2178/jsl/1294171001.

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