Key developments in algorithmic randomness

Abstract

Introduction 1 1.1. Notation 3 1.2. Computability theory 4 1.3. Core mathematical concepts 9 2. Early developments 12 2.1. Randomness via initial segment complexity 13 2.2. Martin-Löf randomness 15 2.3. Schnorr's contributions 17 3. Intermittent work: The late twentieth century 19 3.1. The contributions of Demuth and Kučera 19 3.2. The contributions of Kurtz, Kautz, and van Lambalgen 20 4. Rapid growth at the turn of the century 23 4.1. The Turing degrees of random sequences 23 4.2. Chaitin's Ω 24 4.3. Randomness-theoretic reducibilities 25 4.4. Other randomness notions and lowness for randomness 27 4.5. Effective notions of dimension 35 5. Recent developments 37 6. Acknowledgments 38 References 38