Randomness Through Computation

This review volume consists of an indispensable set of chapters written by leading scholars, scientists and researchers in the field of Randomness, including related subfields specially but not limited to the strong developed connections to the Computability and Recursion Theory. Highly respected, indeed renowned in their areas of specialization, many of these contributors are the founders of their fields. The scope of Randomness Through Computation is novel. Each contributor shares his personal views and anecdotes on the various reasons and motivations which led him to the study of the subject. They share their visions from their vantage and distinctive viewpoints. In summary, this is an opportunity to learn about the topic and its various angles from the leading thinkers.

Proper treatment of structural behavior under severe loading - such as the performance of a high-rise building during an earthquake - relies heavily on the use of probability-based analysis and decision-making tools. Proper application of these tools is significantly enhanced by a thorough understanding of the underlying theoretical and computation

Computability and complexity theory are two central areas of research in theoretical computer science. This book provides a systematic, technical development of "algorithmic randomness" and complexity for scientists from diverse fields.

Surveys on recent developments in the theory of algorithmic randomness and its interactions with other areas of mathematics.

Recent findings in the computer sciences, discrete mathematics, formal logics and metamathematics have opened up a royal road for the investigation of undecidability and randomness in physics. A translation of these formal concepts yields a fresh look into diverse features of physical modelling such as quantum complementarity and the measurement problem, but also stipulates questions related to the necessity of the assumption of continua.Conversely, any computer may be perceived as a physical system: not only in the immediate sense of the physical properties of its hardware. Computers are a medium to virtual realities. The foreseeable importance of such virtual realities stimulates the investigation of an ?inner description?, a ?virtual physics? of these universes of computation. Indeed, one may consider our own universe as just one particular realisation of an enormous number of virtual realities, most of them awaiting discovery.One motive of this book is the recognition that what is often referred to as ?randomness? in physics might actually be a signature of undecidability for systems whose evolution is computable on a step-by-step basis. To give a flavour of the type of questions envisaged: Consider an arbitrary algorithmic system which is computable on a step-by-step basis. Then it is in general impossible to specify a second algorithmic procedure, including itself, which, by experimental input-output analysis, is capable of finding the deterministic law of the first system. But even if such a law is specified beforehand, it is in general impossible to predict the system behaviour in the ?distant future?. In other words: no ?speedup? or ?computational shortcut? is available. In this approach, classical paradoxes can be formally translated into no-go theorems concerning intrinsic physical perception.It is suggested that complementarity can be modelled by experiments on finite automata, where measurements of one observable of the automaton destroys the possibility to measure another observable of the same automaton and it vice versa.Besides undecidability, a great part of the book is dedicated to a formal definition of randomness and entropy measures based on algorithmic information theory.

This book covers pseudorandom number generation algorithms, evaluation techniques, and offers practical advice and code examples. Random Numbers and Computers is an essential introduction or refresher on pseudorandom numbers in computer science. The first comprehensive book on the topic, readers are provided with a practical introduction to the techniques of pseudorandom number generation, including how the algorithms work and how to test the output to decide if it is suitable for a particular purpose. Practical applications are demonstrated with hands-on presentation and descriptions that readers can apply directly to their own work. Examples are in C and Python and given with an emphasis on understanding the algorithms to the point of practical application. The examples are meant to be implemented, experimented with and improved/adapted by the reader.

This is a book of elementary probability theory that includes a chapter on algorithmic randomness. It rigorously presents definitions and theorems in computation theory, and explains the meanings of the theorems by comparing them with mechanisms of the computer, which is very effective in the current computer age. Random number topics have not been treated by any books on probability theory, only some books on computation theory. However, the notion of random number is necessary for understanding the essential relation between probability and randomness. The field of probability has changed very much, thus this book will make and leave a big impact even to expert probabilists. Readers from applied sciences will benefit from this book because it presents a very proper foundation of the Monte Carlo method with practical solutions, keeping the technical level no higher than 1st year university calculus. Contents: Mathematics of Coin TossingMathematical ModelRandom NumberLimit TheoremMonte Carlo MethodInfinite coin TossesRandom Number: Recursive FunctionKolmogorov Complexity and Random NumberLimit Theorem: Bernoulli's TheoremLaw of Large NumbersDe Moivre–Laplace's TheoremCentral Limit TheoremMathematical StatisticsMonte Carlo Method: Monte Carlo Method as GamblingPseudorandom GeneratorMonte Carlo IntegrationFrom the Viewpoint of Mathematical StatisticsAppendices: Symbols and TermsBinary Numeral SystemLimit of Sequence and FunctionLimits of Exponential Function and LogarithmC Language Program Readership: First year university students to professionals. Keywords: Probability;Probability Theory;Randomness;Random Number;Pseudorandom Number;Monte Carlo Method;Monte Carlo IntegrationReview: Key Features: This is the first book that presents both probability theory and algorithmic randomness for from 1st year university students to experts. It is technically easy but worth reading for experts as wellThis book presents basic limit theorems with proofs that are not seen in usual probability textbooks; for readers should learn that a good solution is not always uniqueThis book rigorously treats the Monte Carlo method. In particular, it presents the random Weyl sampling, which produces pseudorandom numbers for the Monte Carlo integration that act complete substitutes for random numbers