The research presented in Aspects of Kolmogorov Complexity addresses the fundamental standard of defining randomness as measured by a Martin-Lof level of randomness as found in random sequential binary strings. A classical study of statistics that addresses both a fundamental standard of statistics as well as an applied measure for statistical communication theory. The research points to compression levels in a random state that are greater than is found in current literature. A historical overview of the field of Kolmogorov Complexity and Algorithmic Information Theory, a subfield of Information Theory, is given as well as examples using a radix 3, radix 4, and radix 5 base numbers for both random and non-random sequential strings. The text also examines monochromatic and chromatic symbols and both theoretical and applied aspects of data compression as they relate to the transmission and storage of information. The appendix contains papers on the subject given at conferences and the references are current.ContentsTechnical topics addressed in Aspects of Kolmogorov Complexity include:• Statistical Communication Theory• Algorithmic Information Theory• Kolmogorov Complexity• Martin-Lof Randomness• Compression, Transmission and Storage of Information

Briefly, we review the basic elements of computability theory and prob ability theory that are required. Finally, in order to place the subject in the appropriate historical and conceptual context we trace the main roots of Kolmogorov complexity. This way the stage is set for Chapters 2 and 3, where we introduce the notion of optimal effective descriptions of objects. The length of such a description (or the number of bits of information in it) is its Kolmogorov complexity. We treat all aspects of the elementary mathematical theory of Kolmogorov complexity. This body of knowledge may be called algo rithmic complexity theory. The theory of Martin-Lof tests for random ness of finite objects and infinite sequences is inextricably intertwined with the theory of Kolmogorov complexity and is completely treated. We also investigate the statistical properties of finite strings with high Kolmogorov complexity. Both of these topics are eminently useful in the applications part of the book. We also investigate the recursion theoretic properties of Kolmogorov complexity (relations with Godel's incompleteness result), and the Kolmogorov complexity version of infor mation theory, which we may call "algorithmic information theory" or "absolute information theory. " The treatment of algorithmic probability theory in Chapter 4 presup poses Sections 1. 6, 1. 11. 2, and Chapter 3 (at least Sections 3. 1 through 3. 4).

The mathematical theory of computation has given rise to two important ap proaches to the informal notion of "complexity": Kolmogorov complexity, usu ally a complexity measure for a single object such as a string, a sequence etc., measures the amount of information necessary to describe the object. Compu tational complexity, usually a complexity measure for a set of objects, measures the compuational resources necessary to recognize or produce elements of the set. The relation between these two complexity measures has been considered for more than two decades, and may interesting and deep observations have been obtained. In March 1990, the Symposium on Theory and Application of Minimal Length Encoding was held at Stanford University as a part of the AAAI 1990 Spring Symposium Series. Some sessions of the symposium were dedicated to Kolmogorov complexity and its relations to the computational complexity the ory, and excellent expository talks were given there. Feeling that, due to the importance of the material, some way should be found to share these talks with researchers in the computer science community, I asked the speakers of those sessions to write survey papers based on their talks in the symposium. In response, five speakers from the sessions contributed the papers which appear in this book.

The latest edition of this classic is updated with new problem sets and material The Second Edition of this fundamental textbook maintains the book's tradition of clear, thought-provoking instruction. Readers are provided once again with an instructive mix of mathematics, physics, statistics, and information theory. All the essential topics in information theory are covered in detail, including entropy, data compression, channel capacity, rate distortion, network information theory, and hypothesis testing. The authors provide readers with a solid understanding of the underlying theory and applications. Problem sets and a telegraphic summary at the end of each chapter further assist readers. The historical notes that follow each chapter recap the main points. The Second Edition features: * Chapters reorganized to improve teaching * 200 new problems * New material on source coding, portfolio theory, and feedback capacity * Updated references Now current and enhanced, the Second Edition of Elements of Information Theory remains the ideal textbook for upper-level undergraduate and graduate courses in electrical engineering, statistics, and telecommunications.

Algorithmic information theory (AIT), or Kolmogorov complexity as it is known to mathematicians, can provide a useful tool for scientists to look at natural systems, however, some critical conceptual issues need to be understood and the advances already made collated and put in a form accessible to scientists. This book has been written in the hope that readers will be able to absorb the key ideas behind AIT so that they are in a better position to access the mathematical developments and to apply the ideas to their own areas of interest. The theoretical underpinning of AIT is outlined in the earlier chapters, while later chapters focus on the applications, drawing attention to the thermodynamic commonality between ordered physical systems such as the alignment of magnetic spins, the maintenance of a laser distant from equilibrium, and ordered living systems such as bacterial systems, an ecology, and an economy. Key Features Presents a mathematically complex subject in language accessible to scientists Provides rich insights into modelling far-from-equilibrium systems Emphasises applications across range of fields, including physics, biology and econophysics Empowers scientists to apply these mathematical tools to their own research

No statistical model is "true" or "false," "right" or "wrong"; the models just have varying performance, which can be assessed. The main theme in this book is to teach modeling based on the principle that the objective is to extract the information from data that can be learned with suggested classes of probability models. The intuitive and fundamental concepts of complexity, learnable information, and noise are formalized, which provides a firm information theoretic foundation for statistical modeling. Although the prerequisites include only basic probability calculus and statistics, a moderate level of mathematical proficiency would be beneficial.

This introduction to the MDL Principle provides a reference accessible to graduate students and researchers in statistics, pattern classification, machine learning, and data mining, to philosophers interested in the foundations of statistics, and to researchers in other applied sciences that involve model selection.