Diagonally non-computable functions and fireworks

TL;DR

This paper demonstrates that the set of diagonally non-computable functions which do not compute Martin-Löf random reals is non-negligible, using advanced forcing techniques and a novel 'fireworks argument'.

Contribution

It proves the non-negligibility of certain DNC functions that do not compute Martin-Löf random reals, extending understanding of their computational complexity.

Findings

DNC functions can be non-negligible despite not computing random reals

For sufficiently fast-growing functions h, every 2-random real computes an h-bounded DNC function

The set of reals computing a DNC but no bounded DNC function is also non-negligible

Abstract

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-L\"of 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-L\"of 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-L\"of 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.