Complexity of Computer Computations

Complexity of Computer Computations PDF

Author: R. Miller

Publisher: Springer Science & Business Media

Published: 2013-03-13

Total Pages: 228

ISBN-13: 1468420011

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The Symposium on the Complexity of Computer Compu tations was held at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York, March 20-22, 1972. These Proceedings contain all papers presented at the Symposium together with a transcript of the concluding panel discussion and a comprehensive bibliography of the field. The Symposium dealt with complexity studies closely re lated to how computations are actually performed on computers. Although this area of study has not yet found an appropriate or generally accepted name, the area is recognizable by the signif icant commonality in problems, approaches, and motivations. The area can be described and delineated by examples such as the following. (1) Determining lower bounds on the number of operations or steps required for computational solutions of specific problems such as matrix and polynomial calculations, sorting and other combinatorial problems, iterative com putations, solving equations, and computer resource allocation. (2) Developing improved algorithms for the solution of such problems which provide good upper bounds on the number of required operations, along with experimental and v vi PREFACE theoretical evidence concerning the efficiency and numer ical accuracy of those algorithms. (3) Studying the effects on the efficiency of computation brought about by variations in sequencing and the intro duction of parallelism.

Complexity of Computer Computations

Complexity of Computer Computations PDF

Author: R. Miller

Publisher: Springer

Published: 1972-08-01

Total Pages: 225

ISBN-13: 9780306307072

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The Symposium on the Complexity of Computer Compu tations was held at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York, March 20-22, 1972. These Proceedings contain all papers presented at the Symposium together with a transcript of the concluding panel discussion and a comprehensive bibliography of the field. The Symposium dealt with complexity studies closely re lated to how computations are actually performed on computers. Although this area of study has not yet found an appropriate or generally accepted name, the area is recognizable by the signif icant commonality in problems, approaches, and motivations. The area can be described and delineated by examples such as the following. (1) Determining lower bounds on the number of operations or steps required for computational solutions of specific problems such as matrix and polynomial calculations, sorting and other combinatorial problems, iterative com putations, solving equations, and computer resource allocation. (2) Developing improved algorithms for the solution of such problems which provide good upper bounds on the number of required operations, along with experimental and v vi PREFACE theoretical evidence concerning the efficiency and numer ical accuracy of those algorithms. (3) Studying the effects on the efficiency of computation brought about by variations in sequencing and the intro duction of parallelism.

Complexity and Real Computation

Complexity and Real Computation PDF

Author: Lenore Blum

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 456

ISBN-13: 1461207010

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The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.

Computational Complexity

Computational Complexity PDF

Author: Sanjeev Arora

Publisher: Cambridge University Press

Published: 2009-04-20

Total Pages: 609

ISBN-13: 0521424267

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New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.

Computational Complexity and Feasibility of Data Processing and Interval Computations

Computational Complexity and Feasibility of Data Processing and Interval Computations PDF

Author: V. Kreinovich

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 460

ISBN-13: 1475727933

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Targeted audience • Specialists in numerical computations, especially in numerical optimiza tion, who are interested in designing algorithms with automatie result ver ification, and who would therefore be interested in knowing how general their algorithms caIi in principle be. • Mathematicians and computer scientists who are interested in the theory 0/ computing and computational complexity, especially computational com plexity of numerical computations. • Students in applied mathematics and computer science who are interested in computational complexity of different numerical methods and in learning general techniques for estimating this computational complexity. The book is written with all explanations and definitions added, so that it can be used as a graduate level textbook. What this book .is about Data processing. In many real-life situations, we are interested in the value of a physical quantity y that is diflicult (or even impossible) to measure directly. For example, it is impossible to directly measure the amount of oil in an oil field or a distance to a star. Since we cannot measure such quantities directly, we measure them indirectly, by measuring some other quantities Xi and using the known relation between y and Xi'S to reconstruct y. The algorithm that transforms the results Xi of measuring Xi into an estimate fj for y is called data processing.

Computational Complexity of Counting and Sampling

Computational Complexity of Counting and Sampling PDF

Author: Istvan Miklos

Publisher: CRC Press

Published: 2019-02-21

Total Pages: 292

ISBN-13: 1351971603

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Computational Complexity of Counting and Sampling provides readers with comprehensive and detailed coverage of the subject of computational complexity. It is primarily geared toward researchers in enumerative combinatorics, discrete mathematics, and theoretical computer science. The book covers the following topics: Counting and sampling problems that are solvable in polynomial running time, including holographic algorithms; #P-complete counting problems; and approximation algorithms for counting and sampling. First, it opens with the basics, such as the theoretical computer science background and dynamic programming algorithms. Later, the book expands its scope to focus on advanced topics, like stochastic approximations of counting discrete mathematical objects and holographic algorithms. After finishing the book, readers will agree that the subject is well covered, as the book starts with the basics and gradually explores the more complex aspects of the topic. Features: Each chapter includes exercises and solutions Ideally written for researchers and scientists Covers all aspects of the topic, beginning with a solid introduction, before shifting to computational complexity’s more advanced features, with a focus on counting and sampling

Mathematics and Computation

Mathematics and Computation PDF

Author: Avi Wigderson

Publisher: Princeton University Press

Published: 2019-10-29

Total Pages: 434

ISBN-13: 0691189137

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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

Computability and Complexity Theory

Computability and Complexity Theory PDF

Author: Steven Homer

Publisher: Springer Science & Business Media

Published: 2011-12-09

Total Pages: 310

ISBN-13: 1461406811

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This revised and extensively expanded edition of Computability and Complexity Theory comprises essential materials that are core knowledge in the theory of computation. The book is self-contained, with a preliminary chapter describing key mathematical concepts and notations. Subsequent chapters move from the qualitative aspects of classical computability theory to the quantitative aspects of complexity theory. Dedicated chapters on undecidability, NP-completeness, and relative computability focus on the limitations of computability and the distinctions between feasible and intractable. Substantial new content in this edition includes: a chapter on nonuniformity studying Boolean circuits, advice classes and the important result of Karp─Lipton. a chapter studying properties of the fundamental probabilistic complexity classes a study of the alternating Turing machine and uniform circuit classes. an introduction of counting classes, proving the famous results of Valiant and Vazirani and of Toda a thorough treatment of the proof that IP is identical to PSPACE With its accessibility and well-devised organization, this text/reference is an excellent resource and guide for those looking to develop a solid grounding in the theory of computing. Beginning graduates, advanced undergraduates, and professionals involved in theoretical computer science, complexity theory, and computability will find the book an essential and practical learning tool. Topics and features: Concise, focused materials cover the most fundamental concepts and results in the field of modern complexity theory, including the theory of NP-completeness, NP-hardness, the polynomial hierarchy, and complete problems for other complexity classes Contains information that otherwise exists only in research literature and presents it in a unified, simplified manner Provides key mathematical background information, including sections on logic and number theory and algebra Supported by numerous exercises and supplementary problems for reinforcement and self-study purposes

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