[Paper Review] A weak randomness notion for probability measures

André Nies, Frank Stephan|arXiv (Cornell University)|Feb 21, 2019

Computability, Logic, AI Algorithms1 citations

TL;DR

This paper introduces a weak randomness notion for probability measures on Cantor space, defined as Martin-Löf absolute continuity, where non-ML-random sequences form a null set under the measure. It establishes connections between this property and the growth rate of initial segment complexity under the measure, showing that maximal complexity growth implies the weak randomness property, though the converse of the Levin-Schnorr theorem fails in this context.

ABSTRACT

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure $\mu$ on the space of infinite bit sequences is ML absolutely continuous if the non-ML-random bit sequences form a null set with respect to~$\mu$. We think of this as a weak randomness notion for measures. We begin with examples, and provide a robustness property related to Solovay tests. Our main work connects our weak randomness notion to the growth of the initial segment complexity for measures~$\mu$; the latter is defined as a $\mu$-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We discuss $C$-triviality and $K$-triviality for measures and relate these two notions with each other. Here triviality means that the growth of initial segment complexity is as slow as possible. We show that full Martin-Lof randomness of a measure implies ML absolute continuity; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures. We conclude with several open questions.

Motivation & Objective

• To define and investigate a weak form of algorithmic randomness for probability measures on Cantor space.
• To explore the robustness of this notion using Solovay tests and its compatibility with measures having atoms.
• To connect the weak randomness property to the average initial segment complexity of strings under the measure.
• To examine $C$-triviality and $K$-triviality for measures, characterizing slow growth of complexity.
• To extend results to weak randomness relative to general ergodic computable measures, seeking effective versions of Shannon-McMillan-Breiman and Brudno theorems.

Proposed method

• Defining ML absolute continuity as the condition that non-Martin-Löf random sequences have measure zero under the given probability measure.
• Using Solovay tests to establish robustness of the weak randomness notion under measure transformations.
• Analyzing the average initial segment complexity of strings of length $n$ under the measure $\mu$, denoted $\mathbb{E}_\mu[K(x)]$.
• Proving that maximal growth of $\mathbb{E}_\mu[K(x)]$ implies ML absolute continuity.
• Demonstrating that the implications of the Levin-Schnorr theorem do not hold in the context of measures.
• Studying $C$-triviality and $K$-triviality for measures as notions of minimal complexity growth.

Experimental results

Research questions

• RQ1 Does ML absolute continuity imply full Martin-Löf randomness for measures?
• RQ2 How does the average initial segment complexity $\mathbb{E}_\mu[K(x)]$ relate to the weak randomness property of $\mu$?
• RQ3 Can the Levin-Schnorr theorem's implications be extended to measures, or do they fail in this setting?
• RQ4 What is the relationship between $C$-triviality and $K$-triviality for measures?
• RQ5 Can effective versions of the Shannon-McMillan-Breiman and Brudno theorems be formulated for measures instead of sequences?

Key findings

• Full Martin-Löf randomness of a measure implies ML absolute continuity, but the converse does not hold due to compatibility with atoms.
• A maximal growth rate of the average initial segment complexity $\mathbb{E}_\mu[K(x)]$ implies ML absolute continuity for the measure $\mu$.
• The implications of the Levin-Schnorr theorem fail in the context of measures, even though the forward direction holds.
• $C$-triviality and $K$-triviality for measures are distinct notions, with $K$-triviality corresponding to the slowest possible growth of complexity.
• ML absolute continuity is robust under Solovay tests, indicating its stability under effective measure transformations.
• The paper identifies open questions regarding effective versions of the Shannon-McMillan-Breiman and Brudno theorems for measures.