Differentiability of polynomial time computable functions

Abstract

We show that a real z is polynomial time random if and only if each nondecreasing polynomial time computable function is differentiable at z. This establishes an analog in feasible analysis of a recent result of Brattka, Miller and Nies, who characterized computable randomness in terms of differentiability of nondecreasing computable functions.

Further, we show that a Martin-Loef random real z is a density-one point if and only if each interval-c.e. function is differentiable at z. (To say z is a density-one point means that every effectively closed class containing z has density one at z. The interval-c.e. functions are, essentially, the variation functions of computable functions.)

The proofs are related: they both make use of the analytical concept of porosity in novel ways, and both use a basic geometric fact on shifting dyadic intervals by 1/3.