Pointed computations and Martin-L\"of randomness

TL;DR

This paper generalizes the characterization of Martin-Löf randomness using incompressibility of initial segments, extending it to pointedly X-computable sequences and exploring the relationship between incompressibility and non-random reals.

Contribution

It establishes a new equivalence for X Martin-Löf randomness using pointedly X-computable sequences, broadening the understanding of randomness characterizations.

Findings

A real is X Martin-Löf random iff initial segments at pointedly X-computable lengths are incompressible.

There exist non-random reals that compute sequences of incompressible initial segments.

The result generalizes previous characterizations of randomness to a broader class of sequences.

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.