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Diagonally non-computable functions and fireworks. (English) Zbl 1423.03141
Summary: A set \(\mathcal{C}\) of reals is said to be negligible if there is no probabilistic algorithm which generates a member of \(\mathcal{C}\) 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 A. Kučera [Lect. Notes Math. 1141, 245–259 (1985; Zbl 0622.03031)] to be non-negligible in a strong sense: every Martin-Löf random real computes a DNC function. K. Ambos-Spies et al. [J. Symb. Log. 69, No. 4, 1089–1104 (2004; Zbl 1076.03039)] 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.

MSC:
03D28 Other Turing degree structures
03D32 Algorithmic randomness and dimension
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