Prompt enumerations and relative randomness
Anthony Morphett · Journal of Logic and Computation · 2010
The ‘dynamic’ property of prompt simplicity has become an influential and important concept in the study of the c.e. Turing degrees. The equivalent property of prompt permitting is a particularly fruitful notion, arising neatly from the technique of Yates permitting. We introduce an analogous notion, prompt non-low-for-randomness. Prompt non-low-for-randomness is a prompt form of non-low-for-random permitting, which is the natural notion of permitting in the context of relative randomness. We begin an investigation of this notion by showing that the class of Turing degrees of promptly non-low-for-random c.e. sets is non-trivial and a proper subclass of the non-low-for-random promptly simple degrees. We also consider prompt non-low-for-randomness in the context of the LR-degrees and LR-reducibility, a weak reducibility arising naturally from relative randomness.