Diagonally non-computable functions and fireworks

A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a non-computable real, the class of coherent completions of Peano Arithmet...

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Bibliographic Details
Published in:Information and computation 2017-04, Vol.253 (Part 1), p.64-77
Main Authors: Bienvenu, Laurent, Patey, Ludovic
Format: Article
Language:English
Subjects:
Computer Science
Diagonally non-computable functions
Fireworks arguments
Levin–V'yugin algebra
Logic in Computer Science
Negligibility
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Summary:A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a non-computable real, the class of coherent completions of Peano Arithmetic or the class of reals of minimal Turing degree. One class of particular interest in the study of negligibility is the class of diagonally non-computable (DNC) functions, proven by Kučera to be non-negligible in a strong sense: every Martin-Löf random real computes a DNC function. Ambos-Spies et al. showed that the converse does not hold: there are DNC functions which compute no Martin-Löf random real. In this paper, we show that the set of such DNC functions is in fact non-negligible using a technique we call ‘fireworks argument’. We also use this technique to prove further results on DNC functions.
ISSN:0890-5401
1090-2651
DOI:10.1016/j.ic.2016.12.008