Algorithmic Randomness and Complexity

Exploring the theory of computability and complexity over the real numbers, this book provides a systematic, technical development of "algorithmic randomness" and complexity for scientists working in diverse fields concerned with theoretical computer science.

Computability and complexity theory are two central areas of research in theoretical computer science. Until recently, most work in these areas concentrated on problems over discrete structures, but there has been enormous growth of computability theory and complexity theory over the real numbers and other continuous structures, especially incorporating concepts of "randomness." This book provides a systematic, technical development of "algorithmic randomness" and complexity. It presents concepts and results for understanding relative randomness and its relation to computational complexity. These new results are important for addressing fundamental problems in computational geometry, modeling of dynamic systems, and classical problems in numerical computations.

Authors were awarded the 2016 Shoenfield Prize by the Association for Symbolic Logic for outstanding expository writing in the field of logic Essential resource for researchers and graduate students in theoretical computer science, logic, computability theory, and complexity First comprehensive treatment on the subject, written by experts Includes supplementary material: sn.pub/extras

Zusammenfassung
The theory of algorithmic randomness uses tools from computability theory and algorithmic information theory to address questions such as these. It will be of interest to researchers and students in computability theory, algorithmic information theory, and theoretical computer science.

Inhalt
Background.- Preliminaries.- Computability Theory.- Kolmogorov Complexity of Finite Strings.- Relating Complexities.- Effective Reals.- Notions of Randomness.- Martin-Löf Randomness.- Other Notions of Algorithmic Randomness.- Algorithmic Randomness and Turing Reducibility.- Relative Randomness.- Measures of Relative Randomness.- Complexity and Relative Randomness for 1-Random Sets.- Randomness-Theoretic Weakness.- Lowness and Triviality for Other Randomness Notions.- Algorithmic Dimension.- Further Topics.- Strong Jump Traceability.- ? as an Operator.- Complexity of Computably Enumerable Sets.