Algorithmic Randomness
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Algorithmic Randomness And Complexity
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Author : Rodney G. Downey
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-10-29
Algorithmic Randomness And Complexity written by Rodney G. Downey and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-29 with Computers categories.
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.
Algorithmic Randomness
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Author : Johanna N. Y. Franklin
language : en
Publisher: Cambridge University Press
Release Date : 2020-05-07
Algorithmic Randomness written by Johanna N. Y. Franklin and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-07 with Computers categories.
Surveys on recent developments in the theory of algorithmic randomness and its interactions with other areas of mathematics.
Kolmogorov Complexity And Algorithmic Randomness
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Author : A. Shen
language : en
Publisher: American Mathematical Society
Release Date : 2022-05-18
Kolmogorov Complexity And Algorithmic Randomness written by A. Shen and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-18 with Mathematics categories.
Looking at a sequence of zeros and ones, we often feel that it is not random, that is, it is not plausible as an outcome of fair coin tossing. Why? The answer is provided by algorithmic information theory: because the sequence is compressible, that is, it has small complexity or, equivalently, can be produced by a short program. This idea, going back to Solomonoff, Kolmogorov, Chaitin, Levin, and others, is now the starting point of algorithmic information theory. The first part of this book is a textbook-style exposition of the basic notions of complexity and randomness; the second part covers some recent work done by participants of the “Kolmogorov seminar” in Moscow (started by Kolmogorov himself in the 1980s) and their colleagues. This book contains numerous exercises (embedded in the text) that will help readers to grasp the material.
Randomness Through Computation
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Author : Hector Zenil
language : en
Publisher: World Scientific
Release Date : 2011
Randomness Through Computation written by Hector Zenil and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Computers categories.
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.
Information And Randomness
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Author : Cristian Calude
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Information And Randomness written by Cristian Calude and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
"Algorithmic information theory (AIT) is the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously", says G.J. Chaitin, one of the fathers of this theory of complexity and randomness, which is also known as Kolmogorov complexity. It is relevant for logic (new light is shed on Gödel's incompleteness results), physics (chaotic motion), biology (how likely is life to appear and evolve?), and metaphysics (how ordered is the universe?). This book, benefiting from the author's research and teaching experience in Algorithmic Information Theory (AIT), should help to make the detailed mathematical techniques of AIT accessible to a much wider audience.
Algorithmic Learning In A Random World
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Author : Vladimir Vovk
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-03-22
Algorithmic Learning In A Random World written by Vladimir Vovk and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-03-22 with Computers categories.
Algorithmic Learning in a Random World describes recent theoretical and experimental developments in building computable approximations to Kolmogorov's algorithmic notion of randomness. Based on these approximations, a new set of machine learning algorithms have been developed that can be used to make predictions and to estimate their confidence and credibility in high-dimensional spaces under the usual assumption that the data are independent and identically distributed (assumption of randomness). Another aim of this unique monograph is to outline some limits of predictions: The approach based on algorithmic theory of randomness allows for the proof of impossibility of prediction in certain situations. The book describes how several important machine learning problems, such as density estimation in high-dimensional spaces, cannot be solved if the only assumption is randomness.
Kolmogorov Complexity And Algorithmic Randomness
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Author : A. Shen
language : en
Publisher:
Release Date : 2017
Kolmogorov Complexity And Algorithmic Randomness written by A. Shen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with MATHEMATICS categories.
Looking at a sequence of zeros and ones, we often feel that it is not random, that is, it is not plausible as an outcome of fair coin tossing. Why? The answer is provided by algorithmic information theory: because the sequence is compressible, that is, it has small complexity or, equivalently, can be produced by a short program. This idea, going back to Solomonoff, Kolmogorov, Chaitin, Levin, and others, is now the starting point of algorithmic information theory. The first part of this book is a textbook-style exposition of the basic notions of complexity and randomness; the second part cover.
Some Results On Algorithmic Randomness And Computability Theoretic Strength
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Author :
language : en
Publisher:
Release Date : 2014
Some Results On Algorithmic Randomness And Computability Theoretic Strength written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.
Algorithmic randomness uses tools from computability theory to give precise formulations for what it means for mathematical objects to be random. When the objects in question are reals (infinite sequences of zeros and ones), it reveals complex interactions between how random they are and how useful they are as computational oracles. The results in this thesis are primarily on interactions of this nature. Chapter 1 provides a brief introduction to notation and basic notions from computability theory. Chapter 2 is on shift-complex sequences, also known as everywhere complex sequences. These are sequences all of whose substrings have uniformly high prefix-free Kolmogorov complexity. Rumyantsev showed that the measure of oracles that compute shift-complex sequences is 0. We refine this result to show that the Martin-Löf random sequences that compute shift-complex sequences compute the halting problem. In the other direction, we answer the question of whether every Martin-Löf random sequence computes a shift-complex sequence in the negative by translating it into a question about diagonally noncomputable (or DNC) functions. The key in this result is analyzing how growth rates of DNC functions affect what they can compute. This is the subject of Chapter 3. Using bushy-tree forcing, we show (with J. Miller) that there are arbitrarily slow-growing (but unbounded) DNC functions that fail to compute a Kurtz random sequence. We also extend Kumabe's result that there is a DNC function of minimal Turing degree by showing that for every oracle X, there is a function f that is DNC relative to X and of minimal Turing degree. Chapter 4 is on how "effective" Lebesgue density interacts with computability-theoretic strength and randomness. Bienvenu, Hölzl, Miller, and Nies showed that if we restrict our attention to the Martin-Löf random sequences, then the positive density sequences are exactly the ones that do not compute the halting problem. We prove several facts around this theorem. For example, one direction of the theorem fails without the assumption of Martin-Löf randomness: Given any sequence X, there is a density-one sequence Y that computes it. Another question we answer is whether a positive density point can have minimal degree. It turns out that every such point is either Martin-Löf random, or computes a 1-generic. In either case, it is nonminimal.
Information Randomness Incompleteness
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Author : Gregory J. Chaitin
language : en
Publisher: World Scientific
Release Date : 1987
Information Randomness Incompleteness written by Gregory J. Chaitin and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Computers categories.
The papers gathered in this book were published over a period of more than twenty years in widely scattered journals. They led to the discovery of randomness in arithmetic which was presented in the recently published monograph on ?Algorithmic Information Theory? by the author. There the strongest possible version of Gdel's incompleteness theorem, using an information-theoretic approach based on the size of computer programs, was discussed. The present book is intended as a companion volume to the monograph and it will serve as a stimulus for work on complexity, randomness and unpredictability, in physics and biology as well as in metamathematics.
Algorithmic Randomness And Kolmogorov Complexity For Qubits
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Author : Tejas Shekhar Bhojraj
language : en
Publisher:
Release Date : 2021
Algorithmic Randomness And Kolmogorov Complexity For Qubits written by Tejas Shekhar Bhojraj and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with categories.
This work extends the theories of algorithmic randomness and Kolmogorov complexity of bitstrings to the quantum realm. Nies and Scholz defined quantum Martin-Löf randomness (q-MLR): the first notion of algorithmic randomness to be defined for infinite sequences of qubits, which are called states. We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum Schnorr randomness is then introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states. We next turn to a quantum analogue of Kolmogorov complexity. We introduce quantum-K (QK), a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of QK. Many properties enjoyed by prefix-free Kolmogorov complexity (K) have analogous versions for QK; notably a counting condition. Several connections between Solovay randomness and (K), including the Chaitin type characterization of Solovay randomness, carry over to those between weak Solovay randomness and QK. Schnorr randomness has a Levin\textendash Schnorr characterization using KcC; a version of K defined using an arbitrary computable measure machine, C. We similarly define QKc, a version of QK. Quantum Schnorr randomness is shown to have a Levin\textendash Schnorr and a Chaitin type characterization using QKc. We then show how classical randomness can be generated from a computable, non-quantum random state. We formalize how `measurement' of a state induces a probability measure on the space of infinite bitstrings. A state is `measurement random' (mR) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-Löf randoms. I.e., measuring a mR state produces a Martin-Löf random bitstring with probability one. While quantum-Martin-Löf random states are mR, we show that the converse fails by defining a computable mR state p which is not quantum-Martin-Löf random. In fact, something stronger is true. Measuring p in any computable basis yields an arithmetically random sequence with probability one. The work concludes by studying the asymptotic von Neumann entropy of computable states.