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Randomness and differentiability

2015; American Mathematical Society; Volume: 368; Issue: 1 Linguagem: Inglês

10.1090/tran/6484

1088-6850

Vasco Brattka, Joseph S. Miller, André Nies,

Cellular Automata and Applications

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z ∈ [ 0 , 1 ] z\in [0,1] is computably random if and only if each nondecreasing computable function [ 0 , 1 ] → R [0,1]\rightarrow \mathbb {R} is differentiable at z z . (2) We prove that a real number z ∈ [ 0 , 1 ] z\in [0,1] is weakly 2-random if and only if each almost everywhere differentiable computable function [ 0 , 1 ] → R [0,1]\rightarrow \mathbb {R} is differentiable at z z . (3) Recasting in classical language results dating from 1975 of the constructivist Demuth, we show that a real z z is Martin-Löf random if and only if every computable function of bounded variation is differentiable at z z , and similarly for absolutely continuous functions. We also use our analytic methods to show that computable randomness of a real is base invariant and to derive other preservation results for randomness notions.