Can an individual sequence of zeros and ones be random?

Abstract

CONTENTS Introduction Chapter I. The main notions and facts § 1.1. The notion of randomness depends on a given probability distribution § 1.2. Three faces of randomness: stochasticness, chaoticness, typicalness § 1.3. Typical, chaotic and stochastic sequences: ways to a mathematical definition 1.3.1. Typicalness 1.3.2. Chaoticness 1.3.3. Stochasticness 1.3.4. Comments § 1.4. Typical and chaotic sequences: basic definitions (for the case of the uniform Bernoulli distribution) 1.4.1. Typicalness 1.4.2. Chaoticness Chapter II. Effectively null sets, constructive support, and typical sequences § 2.1. Effectively null sets, computable distributions, and the statement of Martin-Löf's theorem § 2.2. Proof of Martin-Löf's theorem § 2.3. Different versions of the definition of the notion of typicalness 2.3.1. Schorr's definition of typicalness 2.3.2. Solovay's criterion for typicalness 2.3.3. The axiomatic approach to the definition of typicalness Chapter III. Complexity, entropy, and chaotic sequences § 3.1. Computable mappings § 3.2. Kolmogorov's theorem. Monotone entropy § 3.3. Chaotic sequences Chapter IV. What is a random sequence? § 4.1. The proof of the Levin-Schorr theorem for the uniform Bernoulli distribution § 4.2. The case of an arbitrary probability distribution § 4.3. The proofs of the lemmas Chapter V. Probabilistic machines, a priori probability, and randomness § 5.1. Probabilistic machines § 5.2. A priori probability § 5.3. A priori probability and entropy § 5.4. A priori probability and randomness Chapter VI. The frequency approach to the definition of a random sequence § 6.1. Von Mises' approach. The Church and Kolmogorov-Loveland definitions § 6.2. Relations between different definitions. Ville's construction. Muchnik's theorem. Lambalgen's example 6.2.1. Relations between different definitions 6.2.2. Ville's example 6.2.3. Muchnik's theorem 6.2.4. Lambalgen's example § 6.3. A game-theoretic criterion for typicalness Addendum. A timid criticism regarding probability theory References