Pointed computations and Martin-L\"of randomness

Abstract

Schnorr showed that a real is Martin-Loef random if and only if all of its initial segments are incompressible with respect to prefix-free complexity. Fortnow and independently Nies, Stephan and Terwijn noticed that this statement remains true if we can merely require that the initial segments of the real corresponding to a computable increasing sequence of lengths are incompressible. The purpose of this note is to establish the following generalization of this fact. We show that a real is X Martin-Loef random if and only if its initial segments corresponding to a pointedly X-computable sequence (r_n) (where r_n is computable from X in a self-delimiting way, so that at most the first r_n bits of X are queried in the computation) of lengths are incompressible. On the other hand we also show that there are reals which are very far from being Martin-Loef random, yet they compute an increasing sequence of lengths at which their initial segments are incompressible.