Randomness and Differentiability

Details Randomness and Differentiability Randomness and Differentiability

2011-04-22

arxiv.org/abs/1104.4465v3

Mathematics

Logic

37 pages

Scientific paper

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) We show that a real number z in [0,1] is computably random if and only if every nondecreasing computable function [0,1]->R is differentiable at z. (2) A real number z in [0,1] is weakly 2-random if and only if every almost everywhere differentiable computable function [0,1]->R is differentiable at z. (3) Recasting results of the constructivist Demuth (1975) in classical language, 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 the analytic methods to show that computable randomness of a real is base invariant, and to derive preservation results for randomness notions.

Affiliated with

Also associated with

No associations

Rating

Randomness and Differentiability does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Randomness and Differentiability, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Randomness and Differentiability will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-330922