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Computer Arithmetic And Formal Proofs

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Computer Arithmetic and Formal Proofs

Computer Arithmetic and Formal Proofs Book
Author : Sylvie Boldo,Guillaume Melquiond
Publisher : Elsevier
Release : 2017-11-17
ISBN : 0081011709
Language : En, Es, Fr & De

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Book Description :

Floating-point arithmetic is ubiquitous in modern computing, as it is the tool of choice to approximate real numbers. Due to its limited range and precision, its use can become quite involved and potentially lead to numerous failures. One way to greatly increase confidence in floating-point software is by computer-assisted verification of its correctness proofs. This book provides a comprehensive view of how to formally specify and verify tricky floating-point algorithms with the Coq proof assistant. It describes the Flocq formalization of floating-point arithmetic and some methods to automate theorem proofs. It then presents the specification and verification of various algorithms, from error-free transformations to a numerical scheme for a partial differential equation. The examples cover not only mathematical algorithms but also C programs as well as issues related to compilation. Describes the notions of specification and weakest precondition computation and their practical use Shows how to tackle algorithms that extend beyond the realm of simple floating-point arithmetic Includes real analysis and a case study about numerical analysis

Handbook of Floating Point Arithmetic

Handbook of Floating Point Arithmetic Book
Author : Jean-Michel Muller,Nicolas Brunie,Florent de Dinechin,Claude-Pierre Jeannerod,Mioara Joldes,Vincent Lefèvre,Guillaume Melquiond,Nathalie Revol,Serge Torres
Publisher : Birkhäuser
Release : 2018-05-02
ISBN : 3319765264
Language : En, Es, Fr & De

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Book Description :

Floating-point arithmetic is the most widely used way of implementing real-number arithmetic on modern computers. However, making such an arithmetic reliable and portable, yet fast, is a very difficult task. As a result, floating-point arithmetic is far from being exploited to its full potential. This handbook aims to provide a complete overview of modern floating-point arithmetic. So that the techniques presented can be put directly into practice in actual coding or design, they are illustrated, whenever possible, by a corresponding program. The handbook is designed for programmers of numerical applications, compiler designers, programmers of floating-point algorithms, designers of arithmetic operators, and more generally, students and researchers in numerical analysis who wish to better understand a tool used in their daily work and research.

Essential Logic for Computer Science

Essential Logic for Computer Science Book
Author : Rex Page,Ruben Gamboa
Publisher : MIT Press
Release : 2019-01-08
ISBN : 0262039184
Language : En, Es, Fr & De

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Book Description :

An introduction to applying predicate logic to testing and verification of software and digital circuits that focuses on applications rather than theory. Computer scientists use logic for testing and verification of software and digital circuits, but many computer science students study logic only in the context of traditional mathematics, encountering the subject in a few lectures and a handful of problem sets in a discrete math course. This book offers a more substantive and rigorous approach to logic that focuses on applications in computer science. Topics covered include predicate logic, equation-based software, automated testing and theorem proving, and large-scale computation. Formalism is emphasized, and the book employs three formal notations: traditional algebraic formulas of propositional and predicate logic; digital circuit diagrams; and the widely used partially automated theorem prover, ACL2, which provides an accessible introduction to mechanized formalism. For readers who want to see formalization in action, the text presents examples using Proof Pad, a lightweight ACL2 environment. Readers will not become ALC2 experts, but will learn how mechanized logic can benefit software and hardware engineers. In addition, 180 exercises, some of them extremely challenging, offer opportunities for problem solving. There are no prerequisites beyond high school algebra. Programming experience is not required to understand the book's equation-based approach. The book can be used in undergraduate courses in logic for computer science and introduction to computer science and in math courses for computer science students.

Computer Arithmetic

Computer Arithmetic Book
Author : Earl E Swartzlander,Carl E Lemonds
Publisher : World Scientific
Release : 2015-02-12
ISBN : 981465115X
Language : En, Es, Fr & De

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Book Description :

Computer Arithmetic Volume III is a compilation of key papers in computer arithmetic on floating-point arithmetic and design. The intent is to show progress, evolution, and novelty in the area of floating-point arithmetic. This field has made extraordinary progress since the initial software routines on mainframe computers have evolved into hardware implementations in processors spanning a wide range of performance. Nevertheless, these papers pave the way to the understanding of modern day processors design where computer arithmetic are supported by floating-point units. The goal of Volume III is to collect the defining document for floating-point arithmetic and many of the key papers on the implementation of both binary and decimal floating-point arithmetic into a single volume. Although fewer than forty papers are included, their reference lists will direct the interested reader to other excellent work that could not be included here. Volume III is specifically oriented to the needs of designers and users of both general-purpose computers and special-purpose digital processors. The book should also be useful to systems engineers, computer architects, and logic designers. It is also intended to serve as a primary text for a course on floating-point arithmetic, as well as a supplementary text for courses in digital arithmetic and high-speed signal processing. This volume is part of a 3 volume set: Computer Arithmetic Volume I Computer Arithmetic Volume II Computer Arithmetic Volume III The full set is available for sale in a print-only version. Contents:OverviewFloating-Point AdditionFloating-Point MultiplicationRoundingFused Multiply AddFloating-Point DivisionElementary FunctionsDecimal Floating-Point Arithmetic Readership: Graduate students and research professionals interested in computer arithmetic. Key Features:The papers that are included cover the key concepts needed to develop efficient (fast, small and low-power) floating-point processing unitsThe papers include presentations by the initial developers in their own words to better explain the basic techniquesIncludes five papers on decimal floating-point arithmetic, which has been added to the IEEE standardKeywords:Floating-Point Addition;Floating-Point Multiplication;Floating-Point Division;Decimal Floating-Point Arithmetic

NASA Formal Methods

NASA Formal Methods Book
Author : Jyotirmoy V. Deshmukh
Publisher : Springer Nature
Release : 2022-07-07
ISBN : 3031067738
Language : En, Es, Fr & De

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Book Description :

Download NASA Formal Methods book written by Jyotirmoy V. Deshmukh, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Modern Computer Arithmetic

Modern Computer Arithmetic Book
Author : Richard P. Brent,Paul Zimmermann
Publisher : Cambridge University Press
Release : 2010-11-25
ISBN : 1139492284
Language : En, Es, Fr & De

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Book Description :

Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favourite language, while keeping a high-level description and avoiding too low-level or machine-dependent details. The book is intended for anyone interested in the design and implementation of efficient high-precision algorithms for computer arithmetic, and more generally efficient multiple-precision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions to selected exercises are available from the authors.

Knowing Machines

Knowing Machines Book
Author : Donald A. MacKenzie
Publisher : MIT Press
Release : 1998
ISBN : 9780262631884
Language : En, Es, Fr & De

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Book Description :

The essays are tied together by their explorations of connections (primarily among technology, society, and knowledge) and by their general focus on modern "high" technology. They also share an emphasis on the complexity of technological formation and fixation and on the role of belief (especially self-validating belief) in technological change.

Static Analysis

Static Analysis Book
Author : Cezara Drăgoi,Suvam Mukherjee,Kedar Namjoshi
Publisher : Springer Nature
Release : 2021-10-13
ISBN : 3030888061
Language : En, Es, Fr & De

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Book Description :

This book constitutes the refereed proceedings of the 28th International Symposium on Static Analysis, SAS 2021, held in Chicago, IL, USA, in October 2021. The 18 regular and 4 short papers, carefully reviewed and selected from 48 submissions, are presented in this book together with 1-page summaries of the three invited talks. The papers cover topics such as static program analysis, abstract domain, abstract interpretation, automated deduction, debugging techniques, deductive methods, model checking, data science, program optimizations and transformations, program synthesis, program verification, and security analysis.

Intelligent Computer Mathematics

Intelligent Computer Mathematics Book
Author : James H. Davenport,William M. Farmer,Florian Rabe,Josef Urban
Publisher : Springer
Release : 2011-07-18
ISBN : 3642226736
Language : En, Es, Fr & De

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Book Description :

This book constitutes the joint refereed proceedings of three international events, namely the 18th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, Calculemus 2011, the 10th International Conference on Mathematical Knowledge Management, MKM 2011, and a new track on Systems and Projects descriptions that span both the Calculemus and MKM topics, all held in Bertinoro, Italy, in July 2011. All 51 submissions passed through a rigorous review process. A total of 15 papers were submitted to Calculemus, of which 9 were accepted. Systems and Projects track 2011 there have been 12 papers selected out of 14 submissions while MKM 2011 received 22 submissions, of which 9 were accepted for presentation and publication. The events focused on the use of AI techniques within symbolic computation and the application of symbolic computation to AI problem solving; the combination of computer algebra systems and automated deduction systems; and mathematical knowledge management, respectively.

Formal Methods Applications and Technology

Formal Methods  Applications and Technology Book
Author : Lubos Brim,Boudewijn Haverkort,Martin Leucker,Jaco van de Pol
Publisher : Springer
Release : 2007-06-27
ISBN : 3540709525
Language : En, Es, Fr & De

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Book Description :

This book constitutes the thoroughly refereed joint post-proceedings of the two International Workshops on Formal Methods for Industrial Critical Systems, FMICS 2006, and on Parallel and Distributed Methods in Verification, PDMC 2006, held in Bonn, Germany in August 2006 in the course of the 17th International Conference on Concurrency Theory, CONCUR 2006.

Proofs and Computations

Proofs and Computations Book
Author : Helmut Schwichtenberg,Stanley S. Wainer
Publisher : Cambridge University Press
Release : 2011-12-15
ISBN : 1139504169
Language : En, Es, Fr & De

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Book Description :

Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gödel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11–CA0. Ordinal analysis and the (Schwichtenberg–Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Π11–CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.

Computer Arithmetic Scientific Computation and Mathematical Modelling

Computer Arithmetic  Scientific Computation and Mathematical Modelling Book
Author : Edgar W. Kaucher
Publisher : Unknown
Release : 1991
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Computer Arithmetic Scientific Computation and Mathematical Modelling book written by Edgar W. Kaucher, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Proof and Disproof in Formal Logic

Proof and Disproof in Formal Logic Book
Author : Richard Bornat
Publisher : OUP Oxford
Release : 2005-07-21
ISBN : 0191586765
Language : En, Es, Fr & De

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Book Description :

Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. Formal logic allows you to check a logical claim without considering what the claim means. This highly abstracted idea is an essential and practical part of computer science. The idea of a formal system—a collection of rules and axioms which define a universe of logical proofs—is what gives us programming languages and modern-day programming. This book concentrates on using logic as a tool: making and using formal proofs and disproofs of particular logical claims. The logic it uses—natural deduction—is very small and very simple; working with it helps you see how large mathematical universes can be built on small foundations. The book is divided into four parts: · Part I "Basics" gives an introduction to formal logic with a short history of logic and explanations of some technical words. · Part II "Formal syntactic proof" show you how to do calculations in a formal system where you are guided by shapes and never need to think about meaning. Your experiments are aided by Jape, which can operate as both inquisitor and oracle. · Part III "Formal semantic disproof" shows you how to construct mathematical counterexamples to show that proof is impossible. Jape can check the counterexamples you build. · Part IV "Program specification and proof" describes how to apply your logical understanding to a real computer science problem, the accurate description and verification of programs. Jape helps, as far as arithmetic allows. Aimed at undergraduates and graduates in computer science, logic, mathematics, and philosophy, the text includes reference to and exercises based on the computer software package Jape, an interactive teaching and research tool designed and hosted by the author that is freely available on the web.

Computer Hardware Description Languages and their Applications

Computer Hardware Description Languages and their Applications Book
Author : D. Agnew,L. Claesen,R. Camposano
Publisher : Elsevier
Release : 2014-05-21
ISBN : 1483298027
Language : En, Es, Fr & De

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Book Description :

Hardware description languages (HDLs) have established themselves as one of the principal means of designing electronic systems. The interest in and usage of HDLs continues to spread rapidly, driven by the increasing complexity of systems, the growth of HDL-driven synthesis, the research on formal design methods and many other related advances. This research-oriented publication aims to make a strong contribution to further developments in the field. The following topics are explored in depth: BDD-based system design and analysis; system level formal verification; formal reasoning on hardware; languages for protocol specification; VHDL; HDL-based design methods; high level synthesis; and text/graphical HDLs. There are short papers covering advanced design capture and recent work in high level synthesis and formal verification. In addition, several invited presentations on key issues discuss and summarize recent advances in real time system design, automatic verification of sequential circuits and languages for protocol specification.

Computer Arithmetic and Enclosure Methods

Computer Arithmetic and Enclosure Methods Book
Author : Lidiya Atanassova,Jurgen Herzberger,Jürgen Herzberger
Publisher : North Holland
Release : 1992
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Scientists concerned with the interaction between computer arithmetic, programming languages and scientific computing will be particularly interested in this book. It focuses on papers presented at the conference and highlights the increasing impact of SCAN-91 in this area. The volume contains original research and expository articles on the field of interaction between computer arithmetic and its realization in programming languages and the construction and investigation of numerical algorithms for computing guaranteed error-bounds for the solution of various problems in applied mathematics and engineering. Main subjects are: software tools for enclosure algorithms and inclusion methods with the help of interval arithmetic, in particular for the numerical solution of differential equations.

Elementary Functions

Elementary Functions Book
Author : Jean-Michel Muller
Publisher : Birkhäuser
Release : 2016-11-16
ISBN : 1489979832
Language : En, Es, Fr & De

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Book Description :

This textbook presents the concepts and tools necessary to understand, build, and implement algorithms for computing elementary functions (e.g., logarithms, exponentials, and the trigonometric functions). Both hardware- and software-oriented algorithms are included, along with issues related to accurate floating-point implementation. This third edition has been updated and expanded to incorporate the most recent advances in the field, new elementary function algorithms, and function software. After a preliminary chapter that briefly introduces some fundamental concepts of computer arithmetic, such as floating-point arithmetic and redundant number systems, the text is divided into three main parts. Part I considers the computation of elementary functions using algorithms based on polynomial or rational approximations and using table-based methods; the final chapter in this section deals with basic principles of multiple-precision arithmetic. Part II is devoted to a presentation of “shift-and-add” algorithms (hardware-oriented algorithms that use additions and shifts only). Issues related to accuracy, including range reduction, preservation of monotonicity, and correct rounding, as well as some examples of implementation are explored in Part III. Numerous examples of command lines and full programs are provided throughout for various software packages, including Maple, Sollya, and Gappa. New to this edition are an in-depth overview of the IEEE-754-2008 standard for floating-point arithmetic; a section on using double- and triple-word numbers; a presentation of new tools for designing accurate function software; and a section on the Toom-Cook family of multiplication algorithms. The techniques presented in this book will be of interest to implementers of elementary function libraries or circuits and programmers of numerical applications. Additionally, graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find this a useful reference and resource. PRAISE FOR PREVIOUS EDITIONS “[T]his book seems like an essential reference for the experts (which I'm not). More importantly, this is an interesting book for the curious (which I am). In this case, you'll probably learn many interesting things from this book. If you teach numerical analysis or approximation theory, then this book will give you some good examples to discuss in class." — MAA Reviews (Review of Second Edition) "The rich content of ideas sketched or presented in some detail in this book is supplemented by a list of over three hundred references, most of them of 1980 or more recent. The book also contains some relevant typical programs." — Zentralblatt MATH (Review of Second Edition) “I think that the book will be very valuable to students both in numerical analysis and in computer science. I found [it to be] well written and containing much interesting material, most of the time disseminated in specialized papers published in specialized journals difficult to find." — Numerical Algorithms (Review of First Edition)

Formal Hardware Verification

Formal Hardware Verification Book
Author : Thomas Kropf
Publisher : Springer Science & Business Media
Release : 1997-08-27
ISBN : 9783540634751
Language : En, Es, Fr & De

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Book Description :

This state-of-the-art monograph presents a coherent survey of a variety of methods and systems for formal hardware verification. It emphasizes the presentation of approaches that have matured into tools and systems usable for the actual verification of nontrivial circuits. All in all, the book is a representative and well-structured survey on the success and future potential of formal methods in proving the correctness of circuits. The various chapters describe the respective approaches supplying theoretical foundations as well as taking into account the application viewpoint. By applying all methods and systems presented to the same set of IFIP WG10.5 hardware verification examples, a valuable and fair analysis of the strenghts and weaknesses of the various approaches is given.

Algorithms and Architectures for Parallel Processing

Algorithms and Architectures for Parallel Processing Book
Author : Sang-Soo Yeo,Jong Hyuk Park,Laurence Tianruo Yang,Ching-Hsien Hsu
Publisher : Springer
Release : 2010-05-29
ISBN : 3642131190
Language : En, Es, Fr & De

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Book Description :

It is our great pleasure to welcome you to the proceedings of the 10th annual event of the International Conference on Algorithms and Architectures for Parallel Processing (ICA3PP). ICA3PP is recognized as the main regular event covering the many dimensions of parallel algorithms and architectures, encompassing fundamental theoretical - proaches, practical experimental projects, and commercial components and systems. As applications of computing systems have permeated every aspect of daily life, the power of computing systems has become increasingly critical. Therefore, ICA3PP 2010 aimed to permit researchers and practitioners from industry to exchange inf- mation regarding advancements in the state of the art and practice of IT-driven s- vices and applications, as well as to identify emerging research topics and define the future directions of parallel processing. We received a total of 157 submissions this year, showing by both quantity and quality that ICA3PP is a premier conference on parallel processing. In the first stage, all papers submitted were screened for their relevance and general submission - quirements. These manuscripts then underwent a rigorous peer-review process with at least three reviewers per paper. In the end, 47 papers were accepted for presentation and included in the main proceedings, comprising a 30% acceptance rate.

Type Theory and Formal Proof

Type Theory and Formal Proof Book
Author : Rob Nederpelt,Herman Geuvers
Publisher : Cambridge University Press
Release : 2014-11-06
ISBN : 110703650X
Language : En, Es, Fr & De

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Book Description :

A gentle introduction for graduate students and researchers in the art of formalizing mathematics on the basis of type theory.

Formal Verification of Floating Point Hardware Design

Formal Verification of Floating Point Hardware Design Book
Author : David M. Russinoff
Publisher : Springer
Release : 2018-10-13
ISBN : 3319955136
Language : En, Es, Fr & De

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Book Description :

This is the first book to focus on the problem of ensuring the correctness of floating-point hardware designs through mathematical methods. Formal Verification of Floating-Point Hardware Design advances a verification methodology based on a unified theory of register-transfer logic and floating-point arithmetic that has been developed and applied to the formal verification of commercial floating-point units over the course of more than two decades, during which the author was employed by several major microprocessor design companies. The book consists of five parts, the first two of which present a rigorous exposition of the general theory based on the first principles of arithmetic. Part I covers bit vectors and the bit manipulation primitives, integer and fixed-point encodings, and bit-wise logical operations. Part II addresses the properties of floating-point numbers, the formats in which they are encoded as bit vectors, and the various modes of floating-point rounding. In Part III, the theory is extended to the analysis of several algorithms and optimization techniques that are commonly used in commercial implementations of elementary arithmetic operations. As a basis for the formal verification of such implementations, Part IV contains high-level specifications of correctness of the basic arithmetic instructions of several major industry-standard floating-point architectures, including all details pertaining to the handling of exceptional conditions. Part V illustrates the methodology, applying the preceding theory to the comprehensive verification of a state-of-the-art commercial floating-point unit. All of these results have been formalized in the logic of the ACL2 theorem prover and mechanically checked to ensure their correctness. They are presented here, however, in simple conventional mathematical notation. The book presupposes no familiarity with ACL2, logic design, or any mathematics beyond basic high school algebra. It will be of interest to verification engineers as well as arithmetic circuit designers who appreciate the value of a rigorous approach to their art, and is suitable as a graduate text in computer arithmetic.