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• Preface.- I: Introduction.- 1 Introducing Computation Theory.- 2 Introducing the Book.- II: Pillar S: STATE.- 3 Pure State-Based Computational Models.- 4 The Myhill-Nerode Theorem: Implications and Applications.- 5 Online Turing Machines and the Implications of Online Computing.- 6 Pumping: Computational Pigeonholes in Finitary Systems.- 7 Mobility in Computing: An FA Navigates a Mesh.- 8 The Power of Cooperation: Teams of MFAs on a Mesh.- III: Pillar E: ENCODING.- 9 Countability and Uncountability: The Precursors of ENCODING.- 10 Computability Theory.- 11 A Church-Turing Zoo of Computational Models.- 12 Pairing Functions as Encoding Mechanisms.- IV: Pillar N: NONDETERMINISM.- 13 Nondeterminism as Unbounded Parallelism.- 14 Nondeterministic Finite Automata.- 15 Nondeterminism as Unbounded Search.- 16 Complexity Theory.- V: Pillar P: PRESENTATION/SPECIFICATION.- 17 The Elements of Formal Language Theory.- A A Chapter-Long Text on Discrete Mathematics.- B Selected Exercises, by Chapter.- List of ACRONYMS and SYMBOLS.- References.- Index.
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Computation theory is a discipline that uses mathematical concepts and tools to expose the nature of "computation" and to explain a broad range of computational phenomena: Why is it harder to perform some computations than others? Are the differences in difficulty that we observe inherent, or are they artifacts of the way we try to perform the computations? How does one reason about such questions? This unique textbook strives to endow students with conceptual and manipulative tools necessary to make computation theory part of their professional lives. The work achieves this goal by means of three stratagems that set its approach apart from most other texts on the subject. For starters, it develops the necessary mathematical concepts and tools from the concepts' simplest instances, thereby helping students gain operational control over the required mathematics. Secondly, it organizes development of theory around four "pillars, " enabling students to see computational topics that have the same intellectual origins in physical proximity to one another. Finally, the text illustrates the "big ideas" that computation theory is built upon with applications of these ideas within "practical" domains in mathematics, computer science, computer engineering, and even further afield. Suitable for advanced undergraduate students and beginning graduates, this textbook augments the "classical" models that traditionally support courses on computation theory with novel models inspired by "real, modern" computational topics, such as crowd-sourced computing, mobile computing, robotic path planning, and volunteer computing. Arnold L. Rosenberg is Distinguished Univ. Professor Emeritus at University of Massachusetts, Amherst, USA. Lenwood S. Heath is Professor at Virgina Tech, Blacksburg, USA. .
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• 1. Key developments in algorithmic randomness Johanna N. Y. Franklin and Christopher P. Porter
• 2. Algorithmic randomness in ergodic theory Henry Towsner
• 3. Algorithmic randomness and constructive/computable measure theory Jason Rute
• 4. Algorithmic randomness and layerwise computability Mathieu Hoyrup
• 5. Relativization in randomness Johanna N. Y. Franklin
• 6. Aspects of Chaitin's Omega George Barmpalias
• 7. Biased algorithmic randomness Christopher P. Porter
• 8. Higher randomness Benoit Monin
• 9. Resource bounded randomness and its applications Donald M. Stull
• Index.
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The last two decades have seen a wave of exciting new developments in the theory of algorithmic randomness and its applications to other areas of mathematics. This volume surveys much of the recent work that has not been included in published volumes until now. It contains a range of articles on algorithmic randomness and its interactions with closely related topics such as computability theory and computational complexity, as well as wider applications in areas of mathematics including analysis, probability, and ergodic theory. In addition to being an indispensable reference for researchers in algorithmic randomness, the unified view of the theory presented here makes this an excellent entry point for graduate students and other newcomers to the field.
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• Preface
• Conventions and preliminaries
• Introduction
• Part I. Communication: 1. Deterministic protocols
• 2. Rank
• 3. Randomized protocols
• 4. Numbers on foreheads
• 5. Discrepancy
• 6. Information
• 7. Compressing communication
• 8. Lifting
• Part II. Applications: 9. Circuits and proofs
• 10. Memory size
• 11. Data structures
• 12. Extension Complexity of Polytopes
• 13. Distributed computing.
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Communication complexity is the mathematical study of scenarios where several parties need to communicate to achieve a common goal, a situation that naturally appears during computation. This introduction presents the most recent developments in an accessible form, providing the language to unify several disjoint research subareas. Written as a guide for a graduate course on communication complexity, it will interest a broad audience in computer science, from advanced undergraduates to researchers in areas ranging from theory to algorithm design to distributed computing. The first part presents basic theory in a clear and illustrative way, offering beginners an entry into the field. The second part describes applications including circuit complexity, proof complexity, streaming algorithms, extension complexity of polytopes, and distributed computing. Proofs throughout the text use ideas from a wide range of mathematics, including geometry, algebra, and probability. Each chapter contains numerous examples, figures, and exercises to aid understanding.
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• Fractional Packing and Parametric Search Frameworks.- Budget-Constrained Minimum Cost Flows: The Continuous Case.- Budget-Constrained Minimum Cost Flows: The Discrete Case.- Generalized Processing Networks.- Convex Generalized Flows.
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Michael Holzhauser discusses generalizations of well-known network flow and packing problems by additional or modified side constraints. By exploiting the inherent connection between the two problem classes, the author investigates the complexity and approximability of several novel network flow and packing problems and presents combinatorial solution and approximation algorithms.
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• Foundations, Universality & Early Models: Visual Realization of Universal Computation (Harvey Friedman)
• Specification and Computation (Raymond Turner)
• The Many Forms of Amorphous Computational Systems (Jiri Wiedermann)
• Physics, Computation & the Computation of Physics: Computational Realizability in the Real World (Andrej Bauer)
• What is Ultimately Possible in Physics? (Stephen Wolfram)
• The Computable Universe Hypothesis (Matthew Szudzik)
• Computation in Nature & the World: Bacteria, Turing Machines and Hyperbolic Cellular Automata (Maurice Margenstern)
• Computing on Rings (Genaro Martinez & Andy Adamatzky)
• Computation in Unorganized Systems (Christof Teuscher)
• The Quantum & Computation: What is Computation? (How) Does Nature Compute? (David Deutsch)
• Computational Aspects of Quantum Reality (Adan Cabello)
• Self-Reference, Computability, and Quantum Mechanics (Thomas Breuer & Thomas Schulte-Herbrueggen)
• and other papers.
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This volume, with a foreword by Sir Roger Penrose, discusses the foundations of computation in relation to nature. It focuses on two main questions: What is computation? How does nature compute? The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and philosophical implications. The volume provides a state-of-the-art collection of technical papers and non-technical essays, representing a field that assumes information and computation to be key in understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse's "Calculating Space" (the MIT translation), and a panel discussion transcription on the topic, featuring worldwide experts in quantum mechanics, physics, cognition, computation and algorithmic complexity. The volume is dedicated to the memory of Alan M Turing - the inventor of universal computation, on the 100th anniversary of his birth, and is part of the Turing Centenary celebrations.
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• Set Theory Sets-Basic Terms Functions Cardinalities Counting Arguments and Diagonalization
• Languages: Alphabets, Strings, and Languages Alphabets and Strings Operations on Strings Operations on Languages
• Algorithms Computational Problems Decision Problems Traveling Salesman Problem Algorithms: A First Look History Efficiency in Algorithms Counting Steps in an Algorithm Definitions Useful Theorems Properties of O Notation Finding O: Analyzing an Algorithm Best and Average Case Analysis Tractable and Intractable
• Turing Machines Overview The Turing Machine Model Formal Definition of Turing Machine Configurations of Turing Machines Terminology Some Sample Turing Machines Turing Machines: What Should I Be Able to Do?
• Turing-Completeness Other Versions of Turing Machines Turing Machines to Evaluate a Function E numerating Turing Machines The Church-Turing Thesis A Simple Computer Encodings of Turing Machines Universal Turing Machine
• Undecidability Introduction and Overview Self-Reference and Self-Contradiction in Computer Programs Cardinality of the Set of All Languages over an Alphabet Cardinality of the Set of All Turing Machines Construction of the Undecidable Language ACCEPTTM
• Undecidability and Reducibility Undecidable Problems: Other Examples Reducibility Reducibility and Language Properties Reducibility to Show Undecidability Rice's Theorem (a Super-Theorem) Undecidability: What Does It Mean? Post Correspondence Problem Context-Free Grammars
• Classes NP and NP-Complete The Class NP (Nondeterministic Polynomial) Definition of P and NP Polynomial Reducibility Properties Completeness Intractable and Tractable-Once Again A First NP-Complete Problem: Boolean Satisfiability Cook-Levin Theorem: Proof Conclusion
• More NP-Complete Problems Adding Other Problems to the List of Known NP-Complete Problems Reductions to Prove NP-Completeness Graph Problems Vertex Cover: The First Graph Problem Other Graph Problems Hamiltonian Circuit (HC) Eulerian Circuits (an Interesting Problem in P) Three-Dimensional Matching (3DM) Subset Sum Summary and Reprise
• Other Interesting Questions and Classes Introduction Number Problems Complement Classes Open Quest ions Are There Any Problems in NP-P But Not NP-Complete? PSPACE Reachable Configurations NPSPACE = PSPACE A PSPACE Complete Problem Other PSPACE-Complete Problems The Class EXP Space Restrictions Approaches to Hard Problems in Practice Summary
• Bibliography Index
• Exercises appear at the end of each chapter.
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Limits of Computation: An Introduction to the Undecidable and the Intractable offers a gentle introduction to the theory of computational complexity. It explains the difficulties of computation, addressing problems that have no algorithm at all and problems that cannot be solved efficiently. The book enables readers to understand: * What does it mean for a problem to be unsolvable or to be NP-complete? * What is meant by a computation and what is a general model of a computer? * What does it mean for an algorithm to exist and what kinds of problems have no algorithm? * What problems have algorithms but the algorithm may take centuries to finish? Developed from the authors' course on computational complexity theory, the text is suitable for advanced undergraduate and beginning graduate students without a strong background in theoretical computer science. Each chapter presents the fundamentals, examples, complete proofs of theorems, and a wide range of exercises.
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• 1. Prologue
• 2. The Basics
• 3. Insights and Algorithms
• 4. Needles in a Haystack: The class NP
• 5. Who is the Hardest One of All: NP-Completeness
• 6. The Deep Question: P vs. NP
• 7. Memory, Paths and games
• 8. Grand Unified Theory of Computation
• 9. Simply the Best: Optimization
• 10. The Power of Randomness
• 11. Random Walks and Rapid Mixing
• 12. Counting, Sampling, and Statistical Physics
• 13. When Formulas Freeze: Phase Transitions in Computation
• 14. Quantum Computing
• 15. Epilogue
• 16. Appendix: Mathematical Tools.
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Computational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. But this beauty is often buried underneath layers of unnecessary formalism, and exciting recent results like interactive proofs, phase transitions, and quantum computing are usually considered too advanced for the typical student. This book bridges these gaps by explaining the deep ideas of theoretical computer science in a clear and enjoyable fashion, making them accessible to non-computer scientists and to computer scientists who finally want to appreciate their field from a new point of view. The authors start with a lucid and playful explanation of the P vs. NP problem, explaining why it is so fundamental, and so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduate and undergraduate students, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again.
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• PROLEGOMENA.- Mathematical Preliminaries.- STATE.- Online Automata: Exemplars of "State".- Finite Automata and Regular Languages.- Applications of the Myhill-Nerode Theorem.- Enrichment Topics.- ENCODING.- Countability and Uncountability: The Precursors of "Encoding".- Enrichment Topic: "Efficient" Pairing Functions, with Applications.- Computability Theory.- NONDETERMINISM.- Nondeterministic Online Automata.- Nondeterministic FAs.- Nondeterminism in Computability Theory.- Complexity Theory.
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The abstract branch of theoretical computer science known as Computation Theory typically appears in undergraduate academic curricula in a form that obscures both the mathematical concepts that are central to the various components of the theory and the relevance of the theory to the typical student. This regrettable situation is due largely to the thematic tension among three main competing principles for organizing the material in the course. This book is motivated by the belief that a deep understanding of, and operational control over, the few "big" mathematical ideas that underlie Computation Theory is the best way to enable the typical student to assimilate the "big" ideas of Computation Theory into her daily computational life.
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• Contents Preface.
• 1. Preliminaries.
• 2. Abstract complexity theory.
• 3. P, NP, and E.
• 4. Quantum computation.
• 5. One-way functions, pseudo-random generators.
• 6. Optimization problems. A. Tail bounds. Bibliography. Index.
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There has been a common perception that computational complexity is a theory of 'bad news' because its most typical results assert that various real-world and innocent-looking tasks are infeasible. In fact, 'bad news' is a relative term, and, indeed, in some situations (e.g., in cryptography), we want an adversary to not be able to perform a certain task. However, a 'bad news' result does not automatically become useful in such a scenario. For this to happen, its hardness features have to be quantitatively evaluated and shown to manifest extensively. The book undertakes a quantitative analysis of some of the major results in complexity that regard either classes of problems or individual concrete problems. The size of some important classes are studied using resource-bounded topological and measure-theoretical tools. In the case of individual problems, the book studies relevant quantitative attributes such as approximation properties or the number of hard inputs at each length. One chapter is dedicated to abstract complexity theory, an older field which, however, deserves attention because it lays out the foundations of complexity. The other chapters, on the other hand, focus on recent and important developments in complexity. The book presents in a fairly detailed manner concepts that have been at the centre of the main research lines in complexity in the last decade or so, such as: average-complexity, quantum computation, hardness amplification, resource-bounded measure, the relation between one-way functions and pseudo-random generators, the relation between hard predicates and pseudo-random generators, extractors, derandomization of bounded-error probabilistic algorithms, probabilistically checkable proofs, non-approximability of optimization problems, and others. The book should appeal to graduate computer science students, and to researchers who have an interest in computer science theory and need a good understanding of computational complexity, e.g., researchers in algorithms, AI, logic, and other disciplines. It places emphasis on relevant quantitative attributes of important results in complexity. Its coverage is self-contained and accessible to a wide audience. It features large range of important topics including: derandomization techniques, non-approximability of optimization problems, average-case complexity, quantum computation, one-way functions and pseudo-random generators, resource-bounded measure and topology.
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• Week One: Complexity theory: From Godel to Feynman Complexity theory: From Godel to Feynman History and basic concepts Resources, reductions and P vs. NP Probabilistic and quantum computation Complexity classes Space complexity and circuit complexity Oracles and the polynomial time hierarchy Circuit lower bounds "Natural" proofs of lower bounds Bibliography Average case complexity Average case complexity Bibliography Exploring complexity through reductions Introduction PCP theorem and hardness of computing approximate solutions Which problems have strongly exponential complexity? Toda's theorem: $PH\subseteq P^{\ No. P}$ Bibliography Quantum computation Introduction Bipartite quantum systems Quantum circuits and Shor's factoring algorithm Bibliography Lower bounds: Circuit and communication complexity Communication complexity Lower bounds for probabilistic communication complexity Communication complexity and circuit depth Lower bound for directed $st$-connectivity Lower bound for $FORK$ (continued) Bibliography Proof complexity An introduction to proof complexity Lower bounds in proof complexity Automatizability and interpolation The restriction method Other research and open problems Bibliography Randomness in computation Pseudorandomness Preface Computational indistinguishability Pseudorandom generators Pseudorandom functions and concluding remarks Appendix Bibliography Pseudorandomness-Part II Introduction Deterministic simulation of randomized algorithms The Nisan-Wigderson generator Analysis of the Nisan-Wigderson generator Randomness extractors Bibliography Probabilistic proof systems-Part I Interactive proofs Zero-knowledge proofs Suggestions for further reading Bibliography Probabilistically checkable proofs Introduction to PCPs NP-hardness of PCS A couple of digressions Proof composition and the PCP theorem Bibliography.
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Computational complexity theory is a major research area in mathematics and computer science, the goal of which is to set the formal mathematical foundations for efficient computation. There has been significant development in the nature and scope of the field in the last thirty years. It has evolved to encompass a broad variety of computational tasks by a diverse set of computational models, such as randomized, interactive, distributed, and parallel computations. These models can include many computers, which may behave cooperatively or adversarially. Each summer the IAS/Park City Mathematics Institute Graduate Summer School gathers some of the best researchers and educators in the field to present diverse sets of lectures. This volume presents three weeks of lectures given at the Summer School on Computational Complexity Theory. Topics are structured as follows: Week One: Complexity Theory: From Godel to Feynman. This section of the book gives a general introduction to the field, with the main set of lectures describing basic models, techniques, results, and open problems. Week Two: Lower Bounds on Concrete Models. Topics discussed in this section include communication and circuit complexity, arithmetic and algebraic complexity, and proof complexity. Week Three: Randomness in Computation. Lectures are devoted to different notions of pseudorandomness, interactive proof systems and zero knowledge, and probabilistically checkable proofs (PCPs). The volume is recommended for independent study and is suitable for graduate students and researchers interested in computational complexity. Information for our distributors: Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
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• 1. Introduction.-
• 2. Valiant's Algebraic Model of NP-Completeness.-
• 3. Some Complete Families of Polynomials.-
• 4. Cook's versus Valiant's Hypothesis.-
• 5. The Structure of Valiant's Complexity Classes.-
• 6. Fast Evaluation of Representations of General Linear Groups.-
• 7. The Complexity of Immanants.-
• 8. Separation Results and Future Directions.- References.- List of Notations.- Index.
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The theory of NP-completeness is a cornerstone of computational complexity. This monograph provides a thorough and comprehensive treatment of this concept in the framework of algebraic complexity theory. Many of the results presented are new and published for the first time.Topics include: complete treatment of Valiant's algebraic theory of NP-completeness, interrelations with the classical theory as well as the Blum-Shub-Smale model of computation, questions of structural complexity, fast evaluation of representations of general linear groups, and complexity of immanants.The book can be used at the advanced undergraduate or at the beginning graduate level in either mathematics or computer science.
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• UNIFORM COMPLEXITY. Models of Computation and Complexity Classes. NP-Completeness. The Polynomial-Time Hierarchy and Polynomial Space. Structure of NP. NONUNIFORM COMPLEXITY. Decision Trees. Circuit Complexity. Polynomial-Time Isomorphism. PROBABILISTIC COMPLEXITY. Probabilistic Machines and Complexity Classes. Complexity of Counting. Interactive Proof Systems. Probabilistically Checkable Proofs and NP-Hard Optimization Problems. Bibliography. Index.
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A complete treatment of fundamentals and recent advances in complexity theory Complexity theory studies the inherent difficulties of solving algorithmic problems by digital computers. This comprehensive work discusses the major topics in complexity theory, including fundamental topics as well as recent breakthroughs not previously available in book form. Theory of Computational Complexity offers a thorough presentation of the fundamentals of complexity theory, including NP-completeness theory, the polynomial-time hierarchy, relativization, and the application to cryptography. It also examines the theory of nonuniform computational complexity, including the computational models of decision trees and Boolean circuits, and the notion of polynomial-time isomorphism. The theory of probabilistic complexity, which studies complexity issues related to randomized computation as well as interactive proof systems and probabilistically checkable proofs, is also covered. Extraordinary in both its breadth and depth, this volume: Provides complete proofs of recent breakthroughs in complexity theory Presents results in well-defined form with complete proofs and numerous exercises Includes scores of graphs and figures to clarify difficult material An invaluable resource for researchers as well as an important guide for graduate and advanced undergraduate students, Theory of Computational Complexity is destined to become the standard reference in the field.
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• The Parametric Point of View. Parameterized Tractability. The Basic Definitions. Bounded Search and Problem Kernel. Optimization Problem, Approximation Schemes and their Relation with FPT. The Advice View Revisited and LOGSPACE. Automata and Bounded Treewidth. WQO and the Robertson-Seymour Theorems. Miscellaneous Techniques. Parameterized Intractability. Reductions. An Analogue of Cook's Theorem. Other Hardness Results. The W-Hierarchy. Beyond W-Hardness. k-Move games. Provable Intractability: the Class XP. Structural and Other Results. Another Basis. Classical Complexity. The Monotone and Antimonotone Collapses. Parameterized Reducibilities. Appendix. Problem Guide and Compendium. Research Horizons. References. Index.
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This monograph presents an approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language. In doing so, the reader is introduced to new classes of algorithms which may be analysed more precisely than hereto. The authors have made the book as self-contained as possible and a lot of background material is included. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest in this book.
(source: Nielsen Book Data)

• Part I. Fundamentals: 1. Introduction
• 2. Information-based complexity
• 3. Breaking the curse of dimensionality
• Part II. Some Interesting Topics: 4. Very high-dimensional integration and mathematical finance
• 5. Complexity of path integration
• 6. Are ill-posed problems solvable?
• 7. Complexity of nonlinear problems
• 8. What model of computation should be used by scientists? 9. Do impossibility theorems from formal models limit scientific knowledge? 10. Complexity of linear programming
• 11. Complexity of verification
• 12. Complexity of implementation testing
• 13. Noisy information
• 14. Value of information in computation
• 15. Assigning values to mathematical hypotheses
• 16. Open problems
• 17. A brief history of information-based complexity
• Part III. References: 18. A guide to the literature
• Bibliography
• Subject index
• Author index.
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The twin themes of computational complexity and information pervade this book. It starts with an introduction to the computational complexity of continuous mathematical models, that is, information-based complexity. This is then used to illustrate a variety of topics, including breaking the curse of dimensionality, complexity of path integration, solvability of ill-posed problems, the value of information in computation, assigning values to mathematical hypotheses, and new, improved methods for mathematical finance. The style is informal, and the goals are exposition, insight and motivation. A comprehensive bibliography is provided, to which readers are referred for precise statements of results and their proofs. As the first introductory book on the subject it will be invaluable as a guide to the area for the many students and researchers whose disciplines, ranging from physics to finance, are influenced by the computational complexity of continuous problems.
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• From the contents: Efficient algorithms for polynomial manipulation.- The fastest known matrix multiplication algorithms.- Lower bound techniques from algebraic geometry and topology.- Complete treatment of bilinear complexity theory.
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This is the first book to present an up-to-date and self-contained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is well-suited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra.
(source: Nielsen Book Data)

Complexity theory has been a flourishing area of research in the last ten years and currently provides one of the most active subjects for future research problems in computer science. This volume provides a survey of the subject in the form of a collection of articles written by experts that to- gether provide a comprehensive guide to research. The editors' aim has been to provide an accessible description of the current state of complexity theory, and to demonstrate the breadth of techniques and results that make the subject exciting. Thus, papers run the gamut from sublogarithmic space to exponential time and from new combinatorial techniques to interactive proof systems. As a result, researchers in computer science will find this an excellent starting point for study in the subject and a useful source of the key results known.
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• Algorithmics of $NP$-hard problems Algorithmics of propositional satisfiability problems Semantics of S. Yu. Maslov's iterative method Ergodic properties of Maslov's iterative method Anomalous properties of Maslov's iterative method Possible nontraditional methods of establishing unsatisfiability of propositional formulas Dual algorithms in discrete optimization Models, methods, and modes for the synthesis of program schemes Effective calculi as a technique for search reduction Lower bounds of combinatorial complexity for exponential search reduction On a class of polynomial systems of equations following from the formula for total probability and possibilities for eliminating search in solving them S. Maslov's iterative method: 15 years later (freedom of Choice, neural networks, numerical optimization, uncertainty reasoning, and chemical computing)

This collection contains translations of papers on propositional satisfiability and related logical problems which appeared in ""Problemy Sokrashcheniya Perebora"", published in Russian in 1987 by the Scientific Council 'Cybernetics' of the USSR Academy of Sciences. The problems form the nucleus of this intensively developing area.This translation is dedicated to the memory of two remarkable Russian mathematicians, Sergei Maslov and his wife, Nina Maslova. Maslov is known as the originator of the inverse method in automated deduction, which was discovered at the same time as the resolution method of J. A. Robinson and has approximately the same range of applications. In 1981, Maslov proposed an iterative algorithm for propositional satisfiability based on some general ideas of search described in detail in his posthumously published book, ""Theory of Deductive Systems and Its Applications"" (1986; English 1987). This collection contains translations of papers on propositional satisfiability and related logical problems. The papers related to Maslov's iterative method of search reduction play a significant role.
(source: Nielsen Book Data)

• 1. Overview
• 2. Worst case setting
• 3. Average case setting
• 4. Worst-average case setting
• 5. Average-worst case setting
• 6. Asymptotic setting
• Bibliography
• Glossary
• Indices.
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This is the first book in which noisy information is studied in the context of computational complexity, in other words it deals with the computational complexity of mathematical problems for which available information is partial, noisy and priced. The author develops a general theory of computational complexity of continuous problems with noisy information and gives a number of applications; deterministic as well as stochastic noise is considered. He presents optimal algorithms, optimal information, and complexity bounds in different settings: worst case, average case, mixed worst-average and average-worst, and asymptotic. Particular topics include: existence of optimal linear (affine) algorithms, optimality properties of smoothing spline, regularisation and least squares algorithms (with the optimal choice of the smoothing and regularisation parameters), adaption versus nonadaption, relations between different settings. The book integrates the work of researchers over the last decade in such areas as computational complexity, approximation theory and statistics, and includes many new results. Nearly two hundred exercises are supplied with a view to increasing the reader's understanding of the subject. The material is organised in such a way that it can be used either as a textbook for advanced courses, or as a standard reference for professional computer scientists, statisticians, applied mathematicians, engineers, control theorists and economists.
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• I. ALGORITHMS.
• 1. Problems and Algorithms.
• 2. Turing Machines.
• 3. Undecidability. II. LOGIC.
• 1. Boolean Logic.
• 2. First Order Logic.
• 3. Undecidability in Logic. III. P AND NP.
• 1. Relations between Complexity Classes.
• 2. Reductions and Completeness.
• 3. NP-Complete Problems.
• 4. coNP and Function Problems.
• 5. Randomized Computation.
• 6. Cryptography.
• 7. Approximability.
• 8. On P vs. NP. IV. INSIDE P.
• 1. Parallel Computation.
• 2. Logarithmic Space. V. BEYOND NP.
• 1. The Polynomial Hierarchy.
• 2. Computation That Counts.
• 3. Polynomial Space.
• 4. A Glimpse Beyond. 0201530821T04062001.
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This text offers a comprehensive and accessible treatment of the theory of algorithms and complexity - the elegant body of concepts and methods developed by computer scientists over the past 30 years for studying the performance and limitations of computer algorithms. Among topics covered are: reductions and NP-completeness, cryptography and protocols, randomized algorithms, and approximability of optimization problems, circuit complexity, the "structural" aspects of the P=NP question, parallel computation, the polynomial hierarchy, and many others. Several sophisticated and recent results are presented in a rather simple way, while many more are developed in the form of extensive notes, problems, and hints. The book is surprisingly self-contained, in that it develops all necessary mathematical prerequisites from such diverse fields as computability, logic, number theory, combinatorics and probability.
(source: Nielsen Book Data)

• Approximate counting with uniform constant-depth circuits by M. Ajtai On strong separations from $AC^0$ by E. Allender and V. Gore Parallel matching complexity of Ramsey's theorem by J. Beck On algorithms for simple stochastic games by A. Condon Locally random reductions in interactive complexity theory by J. Feigenbaum An application of game-theoretic techniques to cryptography by M. J. Fischer and R. N. Wright Composition of the universal relation by J. Ha stad and A. Wigderson Practical perfect cryptographic security by U. M. Maurer Fair games against an all-powerful adversary by R. Ostrovsky, R. Venkatesan, and M. Yung Factoring integers and computing discrete logarithms via diophantine approximation by C. P. Schnorr A new lower bound theorem for read-only-once branching programs and its applications by J. Simon and M. Szegedy On the E -isomorphism problem by J. Wang.
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This collection of recent papers on computational complexity theory grew out of activities during a special year at DIMACS. With contributions by some of the leading experts in the field, this book is of lasting value in this fast-moving field, providing expositions not found elsewhere. Although aimed primarily at researchers in complexity theory and graduate students in mathematics or computer science, the book is accessible to anyone with an undergraduate education in mathematics or computer science. By touching on some of the major topics in complexity theory, this book sheds light on this burgeoning area of research.
(source: Nielsen Book Data)