Algebraic Complexity Theory

Author : Peter Bürgisser,Michael Clausen,Mohammad A. Shokrollahi
Publisher : Springer Science & Business Media
Page : 630 pages
File Size : 46,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662033388

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Algebraic Complexity Theory by Peter Bürgisser,Michael Clausen,Mohammad A. Shokrollahi Pdf

The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under standing of the intrinsic computational difficulty of problems.

Author : Peter Bürgisser
Publisher : Springer Science & Business Media
Page : 174 pages
File Size : 44,8 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662041796

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Completeness and Reduction in Algebraic Complexity Theory by Peter Bürgisser Pdf

This is a thorough and comprehensive treatment of the theory of NP-completeness in the framework of algebraic complexity theory. Coverage includes 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.

Author : J. M. Landsberg
Publisher : Cambridge University Press
Page : 353 pages
File Size : 48,8 Mb
Release : 2017-09-28
Category : Computers
ISBN : 9781107199231

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Geometry and Complexity Theory by J. M. Landsberg Pdf

This comprehensive introduction to algebraic complexity theory presents new techniques for analyzing P vs NP and matrix multiplication.

Author : Peter Bürgisser,Michael Clausen,Amin Shokrollahi
Publisher : Springer
Page : 618 pages
File Size : 55,7 Mb
Release : 2012-12-22
Category : Mathematics
ISBN : 3662033399

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Algebraic Complexity Theory by Peter Bürgisser,Michael Clausen,Amin Shokrollahi Pdf

The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under standing of the intrinsic computational difficulty of problems.

Author : Satyanarayana V. Lokam
Publisher : Now Publishers Inc
Page : 177 pages
File Size : 55,8 Mb
Release : 2009-07-20
Category : Computers
ISBN : 9781601982421

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Complexity Lower Bounds Using Linear Algebra by Satyanarayana V. Lokam Pdf

We survey several techniques for proving lower bounds in Boolean, algebraic, and communication complexity based on certain linear algebraic approaches. The common theme among these approaches is to study robustness measures of matrix rank that capture the complexity in a given model. Suitably strong lower bounds on such robustness functions of explicit matrices lead to important consequences in the corresponding circuit or communication models. Many of the linear algebraic problems arising from these approaches are independently interesting mathematical challenges.

Author : Lenore Blum,Felipe Cucker,Michael Shub,Steve Smale
Publisher : Springer Science & Business Media
Page : 456 pages
File Size : 47,5 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9781461207016

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Complexity and Real Computation by Lenore Blum,Felipe Cucker,Michael Shub,Steve Smale Pdf

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.

Author : Henri Cohen
Publisher : Springer Science & Business Media
Page : 556 pages
File Size : 54,7 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662029459

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A Course in Computational Algebraic Number Theory by Henri Cohen Pdf

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

Author : John L. Rhodes,Chrystopher L. Nehaniv
Publisher : World Scientific
Page : 293 pages
File Size : 42,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9789812836960

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Applications of Automata Theory and Algebra by John L. Rhodes,Chrystopher L. Nehaniv Pdf

This book was originally written in 1969 by Berkeley mathematician John Rhodes. It is the founding work in what is now called algebraic engineering, an emerging field created by using the unifying scheme of finite state machine models and their complexity to tie together many fields: finite group theory, semigroup theory, automata and sequential machine theory, finite phase space physics, metabolic and evolutionary biology, epistemology, mathematical theory of psychoanalysis, philosophy, and game theory. The author thus introduced a completely original algebraic approach to complexity and the understanding of finite systems. The unpublished manuscript, often referred to as "The Wild Book," became an underground classic, continually requested in manuscript form, and read by many leading researchers in mathematics, complex systems, artificial intelligence, and systems biology. Yet it has never been available in print until now. This first published edition has been edited and updated by Chrystopher Nehaniv for the 21st century. Its novel and rigorous development of the mathematical theory of complexity via algebraic automata theory reveals deep and unexpected connections between algebra (semigroups) and areas of science and engineering. Co-founded by John Rhodes and Kenneth Krohn in 1962, algebraic automata theory has grown into a vibrant area of research, including the complexity of automata, and semigroups and machines from an algebraic viewpoint, and which also touches on infinite groups, and other areas of algebra. This book sets the stage for the application of algebraic automata theory to areas outside mathematics. The material and references have been brought up to date bythe editor as much as possible, yet the book retains its distinct character and the bold yet rigorous style of the author. Included are treatments of topics such as models of time as algebra via semigroup theory; evolution-complexity relations applicable to both ontogeny and evolution; an approach to classification of biological reactions and pathways; the relationships among coordinate systems, symmetry, and conservation principles in physics; discussion of "punctuated equilibrium" (prior to Stephen Jay Gould); games; and applications to psychology, psychoanalysis, epistemology, and the purpose of life. The approach and contents will be of interest to a variety of researchers and students in algebra as well as to the diverse, growing areas of applications of algebra in science and engineering. Moreover, many parts of the book will be intelligible to non-mathematicians, including students and experts from diverse backgrounds.

Author : Sanjeev Arora,Boaz Barak
Publisher : Cambridge University Press
Page : 609 pages
File Size : 40,8 Mb
Release : 2009-04-20
Category : Computers
ISBN : 9780521424264

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Computational Complexity by Sanjeev Arora,Boaz Barak Pdf

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

Author : Z. Wang,S. Xu,T. Gao
Publisher : Springer
Page : 0 pages
File Size : 53,7 Mb
Release : 2014-01-14
Category : Mathematics
ISBN : 9401107963

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Algebraic Systems of Equations and Computational Complexity Theory by Z. Wang,S. Xu,T. Gao Pdf

Author : Zeke Wang,Senlin Xu,Tang'an Gao
Publisher : Unknown
Page : 264 pages
File Size : 49,6 Mb
Release : 1994
Category : Computers
ISBN : UOM:39015033266480

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Algebraic Systems of Equations and Computational Complexity Theory by Zeke Wang,Senlin Xu,Tang'an Gao Pdf

Significant progress has been made during the last 15 years in the solution of nonlinear systems, particularly in computing fixed points, solving systems of nonlinear equations and applications to equilibrium models.

Author : Amir Shpilka,Amir Yehudayoff
Publisher : Now Publishers Inc
Page : 193 pages
File Size : 53,6 Mb
Release : 2010
Category : Computers
ISBN : 9781601984005

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Arithmetic Circuits by Amir Shpilka,Amir Yehudayoff Pdf

A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities, and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.

Author : Jan Krajicek
Publisher : Cambridge University Press
Page : 361 pages
File Size : 52,9 Mb
Release : 1995-11-24
Category : Computers
ISBN : 9780521452052

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Bounded Arithmetic, Propositional Logic and Complexity Theory by Jan Krajicek Pdf

Discusses the deep connections between logic and complexity theory, and lists a number of intriguing open problems.

Author : Rod Downey,Denis R. Hirschfeldt
Publisher : Walter de Gruyter
Page : 181 pages
File Size : 42,7 Mb
Release : 2011-05-02
Category : Mathematics
ISBN : 9783110889178

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Aspects of Complexity by Rod Downey,Denis R. Hirschfeldt Pdf

The book contains 8 detailed expositions of the lectures given at the Kaikoura 2000 Workshop on Computability, Complexity, and Computational Algebra. Topics covered include basic models and questions of complexity theory, the Blum-Shub-Smale model of computation, probability theory applied to algorithmics (randomized alogrithms), parametric complexity, Kolmogorov complexity of finite strings, computational group theory, counting problems, and canonical models of ZFC providing a solution to continuum hypothesis. The text addresses students in computer science or mathematics, and professionals in these areas who seek a complete, but gentle introduction to a wide range of techniques, concepts, and research horizons in the area of computational complexity in a broad sense.

Author : Jaime Gutierrez,Josef Schicho,Martin Weimann
Publisher : Springer
Page : 213 pages
File Size : 47,5 Mb
Release : 2015-01-20
Category : Computers
ISBN : 9783319150819

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Computer Algebra and Polynomials by Jaime Gutierrez,Josef Schicho,Martin Weimann Pdf

Algebra and number theory have always been counted among the most beautiful mathematical areas with deep proofs and elegant results. However, for a long time they were not considered that important in view of the lack of real-life applications. This has dramatically changed: nowadays we find applications of algebra and number theory frequently in our daily life. This book focuses on the theory and algorithms for polynomials over various coefficient domains such as a finite field or ring. The operations on polynomials in the focus are factorization, composition and decomposition, basis computation for modules, etc. Algorithms for such operations on polynomials have always been a central interest in computer algebra, as it combines formal (the variables) and algebraic or numeric (the coefficients) aspects. The papers presented were selected from the Workshop on Computer Algebra and Polynomials, which was held in Linz at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) during November 25-29, 2013, at the occasion of the Special Semester on Applications of Algebra and Number Theory.