Initial segment complexities of randomness notions

Rupert Hölzl,Thorsten Kräling,Frank Stephan,Guohua Wu

doi:10.1016/j.ic.2013.12.002

Abstract

Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of Aʼs initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that a set is 2-random (that is, Martin-Löf random relative to the halting problem K) iff there is no function f such that for all m and all n>f(m) it holds that C(A(0)A(1)…A(n))⩽n−m; before the proof of this equivalence the notion defined via the latter condition was known as Kolmogorov random.In the present work it is shown that characterisations of this style can also be given for other randomness criteria like strong randomness (also known as weak 2-randomness), Kurtz randomness relative to K, Martin-Löf randomness of PA-incomplete sets, and strong Kurtz randomness; here one does not just quantify over all functions f but over functions f of a specific form. For example, A is Martin-Löf random and PA-incomplete iff there is no A-recursive function f such that for all m and all n>f(m) it holds that C(A(0)A(1)…A(n))⩽n−m. The characterisation for strong randomness relates to functions which are the concatenation of an A-recursive function executed after a K-recursive function; this solves an open problem of Nies.In addition to this, characterisations of a similar style are also given for Demuth randomness, weak Demuth randomness and Schnorr randomness relative to K. Although the unrelativised versions of Kurtz randomness and Schnorr randomness do not admit such a characterisation in terms of plain Kolmogorov complexity, Bienvenu and Merkle gave one in terms of Kolmogorov complexity defined by computable machines.

Highlights

Kolmogorov complexity [9, 13] aims to describe when a set is random in an algorithmic way
The first important result in that direction was that Schnorr [19] proved that a set A is Martin-Lof random if and only if for almost all n the prefix free Kolmogorov complexity H(A(0)A(1) . . . A(n)) of the (n + 1)-th initial segment is at least n
The scope of the present paper is to study the notions of randomness beyond Martin-Lof randomness

Summary

Introduction

Kolmogorov complexity [9, 13] aims to describe when a set is random in an algorithmic way. Strong randomness [8, 17] has various nice characterisations, in particular the following: A is strongly random iff A is Martin-Lof random and forms a minimal pair with K with respect to Turing reducibility [4, Footnote 2] For these notions, in order to quantify the degree of non-randomness of a sequence, one studies from which value f (m) onwards all initial segments can be compressed by m bits. A(n)) ≤ n − m for all n > f (m); here f might be an upper bound of the least possible point with this property as one might want to have that f is in a certain Turing degree This idea is quite natural as Kolmogorov random is just the notion of randomness which is defined by the absence of any such f and which coincides with Martin-Lof random relative to K. For the scientific background of this paper, the reader is referred to the usual textbooks on recursion theory [15, 16, 20] and algorithmic randomness [2, 9, 13]

Characterising Strong Randomness

Characterising Demuth Randomness

Characterising Turing-incomplete Martin-Lof random sets

Conclusion and Future Work