Complexity Lower Bounds Using Linear Algebra

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Complexity Lower Bounds Using Linear Algebra

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Author : Satya Lokam
language : en
Publisher: Now Publishers Inc
Release Date : 2009-07-20

Complexity Lower Bounds Using Linear Algebra written by Satya Lokam and has been published by Now Publishers Inc this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-07-20 with Computers categories.

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.

Lower Bounds In Communication Complexity

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Author : Troy Lee
language : en
Publisher: Now Publishers Inc
Release Date : 2009

Lower Bounds In Communication Complexity written by Troy Lee and has been published by Now Publishers Inc this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Computers categories.

The communication complexity of a function f(x, y) measures the number of bits that two players, one who knows x and the other who knows y, must exchange to determine the value f(x, y). Communication complexity is a fundamental measure of complexity of functions. Lower bounds on this measure lead to lower bounds on many other measures of computational complexity. This monograph surveys lower bounds in the field of communication complexity. Our focus is on lower bounds that work by first representing the communication complexity measure in Euclidean space. That is to say, the first step in these lower bound techniques is to find a geometric complexity measure, such as rank or trace norm, that serves as a lower bound to the underlying communication complexity measure. Lower bounds on this geometric complexity measure are then found using algebraic and geometric tools.

Algebraic Complexity Theory

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Author : Peter Bürgisser
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14

Algebraic Complexity Theory written by Peter Bürgisser and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.

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.

Geometry And Complexity Theory

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Author : J. M. Landsberg
language : en
Publisher: Cambridge University Press
Release Date : 2017-09-28

Geometry And Complexity Theory written by J. M. Landsberg and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-28 with Computers categories.

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

Boolean Function Complexity

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Author : Stasys Jukna
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-01-06

Boolean Function Complexity written by Stasys Jukna and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-06 with Mathematics categories.

Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this “complexity Waterloo” that have been discovered over the past several decades, right up to results from the last year or two. Many open problems, marked as Research Problems, are mentioned along the way. The problems are mainly of combinatorial flavor but their solutions could have great consequences in circuit complexity and computer science. The book will be of interest to graduate students and researchers in the fields of computer science and discrete mathematics.

Complexity And Real Computation

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Author : Lenore Blum
language : en
Publisher: Springer Science & Business Media
Release Date : 1998

Complexity And Real Computation written by Lenore Blum and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Computers categories.

The classical theory of computation has been a 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.

Handbook Of Linear Algebra

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Author : Leslie Hogben
language : en
Publisher: CRC Press
Release Date : 2013-11-26

Handbook Of Linear Algebra written by Leslie Hogben and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-26 with Mathematics categories.

With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and

Computational Complexity And Property Testing

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Author : Oded Goldreich
language : en
Publisher: Springer Nature
Release Date : 2020-04-03

Computational Complexity And Property Testing written by Oded Goldreich and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-03 with Computers categories.

This volume contains a collection of studies in the areas of complexity theory and property testing. The 21 pieces of scientific work included were conducted at different times, mostly during the last decade. Although most of these works have been cited in the literature, none of them was formally published before. Within complexity theory the topics include constant-depth Boolean circuits, explicit construction of expander graphs, interactive proof systems, monotone formulae for majority, probabilistically checkable proofs (PCPs), pseudorandomness, worst-case to average-case reductions, and zero-knowledge proofs. Within property testing the topics include distribution testing, linearity testing, lower bounds on the query complexity (of property testing), testing graph properties, and tolerant testing. A common theme in this collection is the interplay between randomness and computation.

Communication Complexity

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Author : Eyal Kushilevitz
language : en
Publisher: Cambridge University Press
Release Date : 1997

Communication Complexity written by Eyal Kushilevitz and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Computers categories.

Surveys the mathematical theory and applications such as computer networks, VLSI circuits, and data structures.

Stacs 2002

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Author : Helmut Alt
language : en
Publisher: Springer
Release Date : 2003-08-03

Stacs 2002 written by Helmut Alt and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-08-03 with Computers categories.

This book constitutes the refereed proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2002, held in Antibes - Juan les Pins, France, in March 2002. The 50 revised full papers presented together with three invited papers were carefully reviewed and selected from a total of 209 submissions. The book offers topical sections on algorithms, current challenges, computational and structural complexity, automata and formal languages, and logic in computer science.