Download or read book Computational Limitations of Small depth Circuits written by Johan Håstad and published by MIT Press (MA). This book was released on 1987 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proving lower bounds on the amount of resources needed to compute specific functions is one of the most active branches of theoretical computer science. Significant progress has been made recently in proving lower bounds in two restricted models of Boolean circuits. One is the model of small depth circuits, and in this book Johan Torkel Hastad has developed very powerful techniques for proving exponential lower bounds on the size of small depth circuits' computing functions. The techniques described in Computational Limitations for Small Depth Circuitscan be used to demonstrate almost optimal lower bounds on the size of small depth circuits computing several different functions, such as parity and majority. The main tool used in the proof of the lower bounds is a lemma, stating that any AND of small fanout OR gates can be converted into an OR of small fanout AND gates with high probability when random values are substituted for the variables. Hastad also applies this tool to relativized complexity, and discusses in great detail the computation of parity and majority in small depth circuits. Contents:Introduction. Small Depth Circuits. Outline of Lower Bound Proofs. Main Lemma. Lower Bounds for Small Depth Circuits. Functions Requiring Depth k to Have Small Circuits. Applications to Relativized Complexity. How Well Can We Compute Parity in Small Depth? Is Majority Harder than Parity? Conclusions. John Hastad is a postdoctoral fellow in the Department of Mathematics at MIT Computational Limitations of Small Depth Circuitsis a winner of the 1986 ACM Doctoral Dissertation Award.
Download or read book Computational Limitations of Small depth Circuits written by Johan Håstad and published by MIT Press (MA). This book was released on 1987 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proving lower bounds on the amount of resources needed to compute specific functions is one of the most active branches of theoretical computer science. Significant progress has been made recently in proving lower bounds in two restricted models of Boolean circuits. One is the model of small depth circuits, and in this book Johan Torkel Hastad has developed very powerful techniques for proving exponential lower bounds on the size of small depth circuits' computing functions. The techniques described in Computational Limitations for Small Depth Circuitscan be used to demonstrate almost optimal lower bounds on the size of small depth circuits computing several different functions, such as parity and majority. The main tool used in the proof of the lower bounds is a lemma, stating that any AND of small fanout OR gates can be converted into an OR of small fanout AND gates with high probability when random values are substituted for the variables. Hastad also applies this tool to relativized complexity, and discusses in great detail the computation of parity and majority in small depth circuits. Contents:Introduction. Small Depth Circuits. Outline of Lower Bound Proofs. Main Lemma. Lower Bounds for Small Depth Circuits. Functions Requiring Depth k to Have Small Circuits. Applications to Relativized Complexity. How Well Can We Compute Parity in Small Depth? Is Majority Harder than Parity? Conclusions. John Hastad is a postdoctoral fellow in the Department of Mathematics at MIT Computational Limitations of Small Depth Circuitsis a winner of the 1986 ACM Doctoral Dissertation Award.
Download or read book On the Power of Small Depth Computation written by Emanuele Viola and published by Now Publishers Inc. This book was released on 2009 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work we discuss selected topics on small-depth computation, presenting a few unpublished proofs along the way. The four sections contain: (1) A unified treatment of the challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over {0, 1}.(2) An unpublished proof that small bounded-depth circuits (AC0) have exponentially small correlation with the parity function. The proof is due to Klivans and Vadhan; it builds upon and simplifies previous ones. (3) Valiant's simulation of log-depth linear-size circuits of fan-in 2 by sub-exponential size circuits of depth 3 and unbounded fan-in. To our knowledge, a proof of this result has never appeared in full. (4) Applebaum, Ishai, and Kushilevitz's cryptography in bounded depth.
Download or read book Computational Complexity Theory written by Steven Rudich and published by American Mathematical Soc.. This book was released on 2004 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational Complexity Theory is the study of how much of a given resource is required to perform the computations that interest us the most. Four decades of fruitful research have produced a rich and subtle theory of the relationship between different resource measures and problems. At the core of the theory are some of the most alluring open problems in mathematics. This book presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on computational complexity. The first week gives a general introduction to the field, including descriptions of the basic mo.
Download or read book Introduction to Circuit Complexity written by Heribert Vollmer and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: An advanced textbook giving a broad, modern view of the computational complexity theory of boolean circuits, with extensive references, for theoretical computer scientists and mathematicians.
Download or read book Limits to Parallel Computation written by Raymond Greenlaw and published by Oxford University Press, USA. This book was released on 1995 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive analysis of the most important topics in parallel computation. It is written so that it may be used as a self-study guide to the field, and researchers in parallel computing will find it a useful reference for many years to come. The first half of the book consists of an introduction to many fundamental issues in parallel computing. The second half provides lists of P-complete- and open problems. These lists will have lasting value to researchers in both industry and academia. The lists of problems, with their corresponding remarks, the thorough index, and the hundreds of references add to the exceptional value of this resource. While the exciting field of parallel computation continues to expand rapidly, this book serves as a guide to research done through 1994 and also describes the fundamental concepts that new workers will need to know in coming years. It is intended for anyone interested in parallel computing, including senior level undergraduate students, graduate students, faculty, and people in industry. As an essential reference, the book will be needed in all academic libraries.
Download or read book Computational Complexity and Property Testing written by Oded Goldreich and published by Springer Nature. This book was released on 2020-04-03 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Computational Complexity Theory written by Juris Hartmanis and published by American Mathematical Soc.. This book was released on 1989 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational complexity theory is the study of the quantitative laws that govern computing. This book contains the proceedings of the AMS Short Course on Computational Complexity Theory, held at the Joint Mathematics Meetings in Atlanta in January 1988.
Download or read book Algorithms and Complexity written by Bozzano G Luisa and published by Elsevier. This book was released on 2014-06-28 with total page 1011 pages. Available in PDF, EPUB and Kindle. Book excerpt: This first part presents chapters on models of computation, complexity theory, data structures, and efficient computation in many recognized sub-disciplines of Theoretical Computer Science.
Download or read book Extremal Combinatorics written by Stasys Jukna and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a concise, up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called the gems of the theory. A wide spectrum of the most powerful combinatorial tools is presented, including methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A thorough discussion of recent applications to computer science illustrates the inherent usefulness of these methods.
Download or read book Advances in Cryptology CRYPTO 97 written by Burton S.Jr. Kaliski and published by Springer Science & Business Media. This book was released on 1997-08-06 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 17th Annual International Cryptology Conference, CRYPTO'97, held in Santa Barbara, California, USA, in August 1997 under the sponsorship of the International Association for Cryptologic Research (IACR). The volume presents 35 revised full papers selected from 160 submissions received. Also included are two invited presentations. The papers are organized in sections on complexity theory, cryptographic primitives, lattice-based cryptography, digital signatures, cryptanalysis of public-key cryptosystems, information theory, elliptic curve implementation, number-theoretic systems, distributed cryptography, hash functions, cryptanalysis of secret-key cryptosystems.
Download or read book Approximation Randomization and Combinatorial Optimization Algorithms and Techniques written by Michel Goemans and published by Springer. This book was released on 2003-05-15 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the joint refereed proceedings of the 4th International Workshop on Approximation Algorithms for Optimization Problems, APPROX 2001 and of the 5th International Workshop on Ranomization and Approximation Techniques in Computer Science, RANDOM 2001, held in Berkeley, California, USA in August 2001. The 26 revised full papers presented were carefully reviewed and selected from a total of 54 submissions. Among the issues addressed are design and analysis of approximation algorithms, inapproximability results, on-line problems, randomization, de-randomization, average-case analysis, approximation classes, randomized complexity theory, scheduling, routing, coloring, partitioning, packing, covering, computational geometry, network design, and applications in various fields.
Download or read book Feasible Mathematics II written by Peter Clote and published by Springer Science & Business Media. This book was released on 2013-03-13 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: Perspicuity is part of proof. If the process by means of which I get a result were not surveyable, I might indeed make a note that this number is what comes out - but what fact is this supposed to confirm for me? I don't know 'what is supposed to come out' . . . . 1 -L. Wittgenstein A feasible computation uses small resources on an abstract computa tion device, such as a 'lUring machine or boolean circuit. Feasible math ematics concerns the study of feasible computations, using combinatorics and logic, as well as the study of feasibly presented mathematical structures such as groups, algebras, and so on. This volume contains contributions to feasible mathematics in three areas: computational complexity theory, proof theory and algebra, with substantial overlap between different fields. In computational complexity theory, the polynomial time hierarchy is characterized without the introduction of runtime bounds by the closure of certain initial functions under safe composition, predicative recursion on notation, and unbounded minimization (S. Bellantoni); an alternative way of looking at NP problems is introduced which focuses on which pa rameters of the problem are the cause of its computational complexity and completeness, density and separation/collapse results are given for a struc ture theory for parametrized problems (R. Downey and M. Fellows); new characterizations of PTIME and LINEAR SPACE are given using predicative recurrence over all finite tiers of certain stratified free algebras (D.
Download or read book Boolean Function Complexity written by Michael S. Paterson and published by Cambridge University Press. This book was released on 1992-11-05 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here Professor Paterson brings together papers from the 1990 Durham symposium on Boolean function complexity. The participants include many well known figures in the field.
Download or read book Theory of Computational Complexity written by Ding-Zhu Du and published by John Wiley & Sons. This book was released on 2014-07-18 with total page 512 pages. Available in PDF, EPUB and Kindle. Book excerpt: Praise for the First Edition "...complete, up-to-date coverage of computational complexitytheory...the book promises to become the standard reference oncomputational complexity." -Zentralblatt MATH A thorough revision based on advances in the field ofcomputational complexity and readers’ feedback, the SecondEdition of Theory of Computational Complexity presentsupdates to the principles and applications essential tounderstanding modern computational complexity theory. The newedition continues to serve as a comprehensive resource on the useof software and computational approaches for solving algorithmicproblems and the related difficulties that can be encountered. Maintaining extensive and detailed coverage, Theory ofComputational Complexity, Second Edition, examines the theoryand methods behind complexity theory, such as computational models,decision tree complexity, circuit complexity, and probabilisticcomplexity. The Second Edition also features recentdevelopments on areas such as NP-completeness theory, as wellas: A new combinatorial proof of the PCP theorem based on thenotion of expander graphs, a research area in the field of computerscience Additional exercises at varying levels of difficulty to furthertest comprehension of the presented material End-of-chapter literature reviews that summarize each topic andoffer additional sources for further study Theory of Computational Complexity, Second Edition, is anexcellent textbook for courses on computational theory andcomplexity at the graduate level. The book is also a usefulreference for practitioners in the fields of computer science,engineering, and mathematics who utilize state-of-the-art softwareand computational methods to conduct research. Athorough revision based on advances in the field of computationalcomplexity and readers’feedback,the Second Edition of Theory of Computational Complexity presentsupdates to theprinciplesand applications essential to understanding modern computationalcomplexitytheory.The new edition continues to serve as a comprehensive resource onthe use of softwareandcomputational approaches for solving algorithmic problems and therelated difficulties thatcanbe encountered.Maintainingextensive and detailed coverage, Theory of ComputationalComplexity, SecondEdition,examines the theory and methods behind complexity theory, such ascomputationalmodels,decision tree complexity, circuit complexity, and probabilisticcomplexity. The SecondEditionalso features recent developments on areas such as NP-completenesstheory, as well as:•A new combinatorial proof of the PCP theorem based on the notion ofexpandergraphs,a research area in the field of computer science•Additional exercises at varying levels of difficulty to furthertest comprehension ofthepresented material•End-of-chapter literature reviews that summarize each topic andoffer additionalsourcesfor further studyTheoryof Computational Complexity, Second Edition, is an excellenttextbook for courses oncomputationaltheory and complexity at the graduate level. The book is also auseful referenceforpractitioners in the fields of computer science, engineering, andmathematics who utilizestate-of-the-artsoftware and computational methods to conduct research.
Download or read book Computational Learning Theory written by David Helmbold and published by Springer Science & Business Media. This book was released on 2001-07-04 with total page 639 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 14th Annual and 5th European Conferences on Computational Learning Theory, COLT/EuroCOLT 2001, held in Amsterdam, The Netherlands, in July 2001. The 40 revised full papers presented together with one invited paper were carefully reviewed and selected from a total of 69 submissions. All current aspects of computational learning and its applications in a variety of fields are addressed.
Download or read book Fundamentals of Computation Theory written by Maciej Liskiewicz and published by Springer. This book was released on 2005-09-09 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is dedicated to the 15th Symposium on Fundamentals of Computation Theory FCT 2005, held in Lubeck, Germany, on August 17–20, 2005.
