Randomness and Differentiability

TL;DR

This paper links algorithmic randomness notions with differentiability properties of effective functions, providing characterizations for computable, weakly 2-random, and ML random reals.

Contribution

It introduces new characterizations of randomness notions via differentiability of computable functions, connecting computability theory with classical analysis.

Findings

Computably random reals are characterized by differentiability of nondecreasing computable functions.

Weakly 2-random reals are characterized by differentiability of almost everywhere differentiable computable functions.

ML random reals are characterized by differentiability of computable functions of bounded variation.

Abstract

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if each nondecreasing computable function [0,1]->R is differentiable at z. (2) We prove that a real number z in [0,1] is weakly 2-random if and only if each almost everywhere differentiable computable function [0,1]->R is differentiable at z. (3) Recasting in classical language results dating from 1975 of the constructivist Demuth, we show that a real z is ML random if and only if every computable function of bounded variation is differentiable at 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.