Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Randomness Through Computation PDF full book. Access full book title Randomness Through Computation by Hector Zenil. Download full books in PDF and EPUB format.
Author: Hector Zenil Publisher: World Scientific ISBN: 9814327743 Category : Computers Languages : en Pages : 439
Book Description
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.
Author: Hector Zenil Publisher: World Scientific ISBN: 9814327743 Category : Computers Languages : en Pages : 439
Book Description
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.
Author: Hector Zenil Publisher: World Scientific ISBN: 9814462632 Category : Computers Languages : en Pages : 439
Book Description
This review volume consists of a set of chapters written by leading scholars, most of them founders of their fields. It explores the connections of Randomness to other areas of scientific knowledge, especially its fruitful relationship to Computability and Complexity Theory, and also to areas such as Probability, Statistics, Information Theory, Biology, Physics, Quantum Mechanics, Learning Theory and Artificial Intelligence. The contributors cover these topics without neglecting important philosophical dimensions, sometimes going beyond the purely technical to formulate age old questions relating to matters such as determinism and free will.The scope of Randomness Through Computation is novel. Each contributor shares their personal views and anecdotes on the various reasons and motivations which led them to the study of Randomness. Using a question and answer format, they share their visions from their several distinctive vantage points.
Author: Publisher: ISBN: Category : Languages : en Pages : 93
Book Description
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.
Author: Edward Barron Publisher: Emereo Publishing ISBN: 9781488866036 Category : Reference Languages : en Pages : 50
Book Description
Finally, a new Randomness Guide. There has never been a Randomness Guide like this. It contains 67 answers, much more than you can imagine; comprehensive answers and extensive details and references, with insights that have never before been offered in print. Get the information you need--fast! This all-embracing guide offers a thorough view of key knowledge and detailed insight. This Guide introduces what you want to know about Randomness. A quick look inside of some of the subjects covered: Applications of randomness - Divination, Applications of randomness - Analysis, Applications of randomness - Science, Randomness - In statistics, Randomness - Generating randomness, Randomness - In the physical sciences, Randomness - A number is due, Kolmogorov randomness - A more formal treatment, Per Martin-Lof - Randomness and Kolmogorov complexity, Random ballot - Randomness in other electoral systems, Randomness - In finance, Randomness - A number is cursed or blessed, Randomness tests - Background, Algorithmic randomness - History, Randomness - Applications and use of randomness, Kolmogorov randomness - Basic results, Representativeness heuristic - Randomness, Retirement Monte Carlo: Better allowance for randomness, Applications of randomness - Simulation, Probability - Relation to randomness, Randomness - In biology, Applications of randomness - Literature, music and art, Algorithmic randomness - Relative randomness, History of randomness, Algorithmic randomness - Properties and examples of Martin-Lof random sequences, Pseudorandomness, Randomization function - Randomness, Pseudorandomness - History, Applications of randomness - Other uses, Applications of randomness - Games, Pseudorandomness - Cryptography, Randomness - Randomness and politics, Fooled by Randomness - Thesis, and much more...
Author: Thomas Weir Watson Publisher: ISBN: Category : Languages : en Pages : 370
Book Description
This dissertation explores the multifaceted interplay between efficient computation and probability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem's output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling through the lens of computational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that "a little more time gives a lot more power to sampling algorithms." Part II concerns settings where the algorithm's output is random. Even when the specification of a computational problem involves no randomness, one can still consider randomized algorithms that produce a correct output with high probability. A basic question is whether one can derandomize algorithms, i.e., reduce or eliminate their dependence on randomness without incurring much loss in efficiency. Although derandomizing arbitrary time-efficient algorithms can only be done assuming unproven complexity-theoretic conjectures, it is possible to unconditionally construct derandomization tools called pseudorandom generators for restricted classes of algorithms. We give an improved pseudorandom generator for a new class, which we call combinatorial checkerboards. The notion of pseudorandomness shows up in many contexts besides derandomization. The so-called Dense Model Theorem, which has its roots in the famous theorem of Green and Tao that the primes contain arbitrarily long arithmetic progressions, is a result about pseudorandomness that has turned out to be a useful tool in computational complexity and cryptography. At the heart of this theorem is a certain type of reduction, and in this dissertation we prove a strong lower bound on the advice complexity of such reductions, which is analogous to a list size lower bound for list-decoding of error-correcting codes. Part III concerns settings where the problem's input is random. We focus on the topic of randomness extraction, which is the problem of transforming a sample from a high-entropy but non-uniform probability distribution (that represents an imperfect physical source of randomness) into a uniformly distributed sample (which would then be suitable for use by a randomized algorithm). It is natural to model the input distribution as being sampled by an efficient algorithm (since randomness is generated by efficient processes in nature), and we give a construction of an extractor for the case where the input distribution's sampler is extremely efficient in parallel time. A related problem is to estimate the min-entropy ("amount of randomness") of a given parallel-time-efficient sampler, since this dictates how many uniformly random output bits one could hope to extract from it. We characterize the complexity of this problem, showing that it is "slightly harder than NP-complete." Part IV concerns settings where the algorithm's input is random. Average-case complexity is the study of the power of algorithms that are allowed to fail with small probability over a randomly chosen input. This topic is motivated by cryptography and by modeling heuristics. A fundamental open question is whether the average-case hardness of NP is implied by the worst-case hardness of NP. We exhibit a new barrier to showing this implication, by proving that a certain general technique (namely, "relativizing proofs by reduction") cannot be used. We also prove results on hardness amplification, clarifying the quantitative relationship between the running time of an algorithm and the probability of failure over a random input (both of which are desirable to minimize).
Author: Avi Wigderson Publisher: Princeton University Press ISBN: 0691189137 Category : Computers Languages : en Pages : 434
Book Description
An introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography
Author: ACM Special Interest Group for Algorithms and Computation Theory Publisher: ISBN: 9781581139600 Category : Computational complexity Languages : en Pages : 798
Author: Hector Zenil
Author: Hector Zenil
Author: Hector Zenil
Author: 
Author: 
Author: Edward Barron
Author: Thomas Weir Watson
Author: 
Author: Avi Wigderson
Author: 
Author: ACM Special Interest Group for Algorithms and Computation Theory