Download or read book Randomness Through Computation written by Hector Zenil and published by World Scientific. This book was released on 2011 with total page 439 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Randomness Through Computation Some Answers More Questions written by Hector Zenil and published by World Scientific. This book was released on 2011-02-11 with total page 439 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Some Results on Algorithmic Randomness and Computability theoretic Strength written by and published by . This book was released on 2014 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Randomness in Physics written by and published by . This book was released on 2009 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Randomness 67 Success Secrets 67 Most Asked Questions on Randomness What You Need to Know written by Edward Barron and published by Emereo Publishing. This book was released on 2014-10-30 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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...
Download or read book The Computational Complexity of Randomness written by Thomas Weir Watson and published by . This book was released on 2013 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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).
Download or read book The Bulletin of Symbolic Logic written by and published by . This book was released on 2006 with total page 720 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematics and Computation written by Avi Wigderson and published by Princeton University Press. This book was released on 2019-10-29 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book Proceedings of the Twenty seventh Annual ACM Symposium on Theory of Computing written by and published by . This book was released on 1995 with total page 780 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book STOC 05 written by ACM Special Interest Group for Algorithms and Computation Theory and published by . This book was released on 2005 with total page 798 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Computational Complexity written by Sanjeev Arora and published by Cambridge University Press. This book was released on 2009-04-20 with total page 609 pages. Available in PDF, EPUB and Kindle. Book excerpt: New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.
Download or read book Foundations of Software Technology and Theoretical Computer Science written by and published by . This book was released on 1999 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Some Applications of Randomness in Computational Complexity written by Luke Friedman and published by . This book was released on 2013 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation we consider two different notions of randomness and their applications to problems in complexity theory. In part one of the dissertation we consider Kolmogorov complexity, a way to formalize a measure of the randomness of a single finite string, something that cannot be done using the usual distributional definitions. We let R be the set of random strings under this measure and study what resource-bounded machines can compute using R as an oracle. We show the surprising result that under proper definitions we can in fact define well-formed complexity classes using this approach, and that perhaps it is possible to exactly characterize standard classes such as BPP and NEXP in this way. In part two of the dissertation we switch gears and consider the use of randomness as a tool in propositional proof complexity, a sub-area of complexity theory that addresses the NP vs. coNP problem. Here we consider the ability of various proof systems to efficiently refute randomly generated unsatisfiable 3-CNF and 3-XOR formulas. In particular, we show that certain restricted proof systems based on Ordered Binary Decision Diagrams requires exponential-size refutations of these formulas. We also outline a new general approach for proving proof complexity lower bounds using random 3-CNF formulas and demonstrate its use on treelike resolution, a weak proof system.
Download or read book Randomness and Computation written by Silvio Micali and published by JAI Press(NY). This book was released on 1989 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Proceedings of the Thirty eighth Annual ACM Symposium on Theory of Computing written by ACM Special Interest Group for Algorithms and Computation Theory and published by . This book was released on 2006 with total page 790 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Kolmogorov Complexity and Algorithmic Randomness written by A. Shen and published by American Mathematical Soc.. This book was released on 2017-11-02 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Computability and Randomness written by André Nies and published by OUP Oxford. This book was released on 2012-03-29 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. The book covers topics such as lowness and highness properties, Kolmogorov complexity, betting strategies and higher computability. Both the basics and recent research results are desribed, providing a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.
