2014/11/25 by Laurent Bienvenu, Ludovic Patey, Bienvenu, Laurent +1
Computer Science · Mathematics · #03B30 03F35 #Advanced Topology and Set Theory #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.1411.6846
openalex publication_date 2014/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 degrees. One class of particular interest in the study of negligibility is the class of diagonally non-computable (DNC) functions, proven by Kucera 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 such the set of such DNC functions is in fact non-negligible. More precisely, we prove that for every sufficiently fast-growing computable~h, every 2-random real computes an h-bounded DNC function which computes no Martin-Löf random real. Further, we show that the same holds for the set of reals which compute a DNC function but no bounded DNC function. The proofs of these results use a combination of a technique due to Kautz (which, following a metaphor of Shen, we like to call a `fireworks argument') and bushy tree forcing, which is the canonical forcing notion used in the study of DNC functions.