Diagonally non-computable functions and fireworks

作者：Bienvenu Laurent^*; Patey Ludovic

来源：Information and Computation, 2017, 253: 64-77.

DOI：10.1016/j.ic.2016.12.008

• Levin-V'yugin algebra
• Negligibility
• Diagonally non-computable functions
• Fireworks arguments

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 Kutera to be non-negligible in a strong sense: every Martin-Lof 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-LOf 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.

• 出版日期2017-4