Computer Arithmetic And Formal Proofs

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Computer Arithmetic and Formal Proofs

Sylvie Boldo,Guillaume Melquiond
Computer Arithmetic and Formal Proofs Book PDF

Author : Sylvie Boldo,Guillaume Melquiond
Publisher : Elsevier
Page : 326 pages
File Size : 40,6 Mb
Release : 2017-11-17
Category : Computers
ISBN : 9780081011706

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Computer Arithmetic and Formal Proofs by Sylvie Boldo,Guillaume Melquiond Pdf

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

Proof and Disproof in Formal Logic

Richard Bornat
Proof and Disproof in Formal Logic Book PDF

Author : Richard Bornat
Publisher : OUP Oxford
Page : 264 pages
File Size : 54,9 Mb
Release : 2005-07-21
Category : Mathematics
ISBN : 9780191586767

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Proof and Disproof in Formal Logic by Richard Bornat Pdf

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 Arithmetic and Validity

Ulrich Kulisch
Computer Arithmetic and Validity Book PDF

Author : Ulrich Kulisch
Publisher : Walter de Gruyter
Page : 456 pages
File Size : 54,7 Mb
Release : 2013-04-30
Category : Mathematics
ISBN : 9783110301793

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Computer Arithmetic and Validity by Ulrich Kulisch Pdf

This is the revised and extended second edition of the successful basic book on computer arithmetic. It is consistent with the newest recent standard developments in the field. The book shows how the arithmetic and mathematical capability of the digital computer can be enhanced in a quite natural way. The work is motivated by the desire and the need to improve the accuracy of numerical computing and to control the quality of the computed results (validity). The accuracy requirements for the elementary floating-point operations are extended to the customary product spaces of computations including interval spaces. The mathematical properties of these models are extracted into an axiomatic approach which leads to a general theory of computer arithmetic. Detailed methods and circuits for the implementation of this advanced computer arithmetic on digital computers are developed in part two of the book. Part three then illustrates by a number of sample applications how this extended computer arithmetic can be used to compute highly accurate and mathematically verified results. The book can be used as a high-level undergraduate textbook but also as reference work for research in computer arithmetic and applied mathematics.

Proofs and Computations

Helmut Schwichtenberg,Stanley S. Wainer
Proofs and Computations Book PDF

Author : Helmut Schwichtenberg,Stanley S. Wainer
Publisher : Cambridge University Press
Page : 480 pages
File Size : 48,6 Mb
Release : 2011-12-15
Category : Mathematics
ISBN : 0521517699

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Proofs and Computations by Helmut Schwichtenberg,Stanley S. Wainer Pdf

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 in Theory and Practice

Ulrich W. Kulisch,Willard L. Miranker
Computer Arithmetic in Theory and Practice Book PDF

Author : Ulrich W. Kulisch,Willard L. Miranker
Publisher : Academic Press
Page : 268 pages
File Size : 42,5 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9781483264738

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Computer Arithmetic in Theory and Practice by Ulrich W. Kulisch,Willard L. Miranker Pdf

Computer Arithmetic in Theory and Practice deals with computer arithmetic and the various implementations of the entire arithmetic package on diverse processors, including microprocessors. It illustrates the importance of theoretical development in the sound implementation of arithmetic on computers, and argues that such an implementation requires the establishment of various isomorphisms between different definitions of arithmetic operations. Comprised of seven chapters, this volume begins with an introduction to the theory of computer arithmetic by giving an axiomatic characterization of the essential properties of sets and subsets; complete lattices and complete subnets; screens and roundings; and arithmetic operations. The discussion then turns to the concepts of a ringoid and a vectoid as well as those of ordered or weakly ordered ringoids and vectoids; interval arithmetic; and floating-point arithmetic. The operations in interval spaces are defined by means of semimorphisms. The final chapter shows how to embed the five basic data types (integer, real, complex, real interval, and complex interval) together with the arithmetic operations that are defined for all of these types into existing higher programming languages. This book will be helpful to students and practitioners in the fields of computer science and applied mathematics.

Concepts of Proof in Mathematics, Philosophy, and Computer Science

Dieter Probst,Peter Schuster
Concepts of Proof in Mathematics Philosophy and Computer Science Book PDF

Author : Dieter Probst,Peter Schuster
Publisher : Walter de Gruyter GmbH & Co KG
Page : 384 pages
File Size : 47,6 Mb
Release : 2016-07-25
Category : Philosophy
ISBN : 9781501502644

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Concepts of Proof in Mathematics, Philosophy, and Computer Science by Dieter Probst,Peter Schuster Pdf

A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

Handbook of Floating-Point Arithmetic

Jean-Michel Muller,Nicolas Brunie,Florent de Dinechin,Claude-Pierre Jeannerod,Mioara Joldes,Vincent Lefèvre,Guillaume Melquiond,Nathalie Revol,Serge Torres
Handbook of Floating Point Arithmetic Book PDF

Author : Jean-Michel Muller,Nicolas Brunie,Florent de Dinechin,Claude-Pierre Jeannerod,Mioara Joldes,Vincent Lefèvre,Guillaume Melquiond,Nathalie Revol,Serge Torres
Publisher : Birkhäuser
Page : 627 pages
File Size : 53,7 Mb
Release : 2018-05-02
Category : Mathematics
ISBN : 9783319765266

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Handbook of Floating-Point Arithmetic by Jean-Michel Muller,Nicolas Brunie,Florent de Dinechin,Claude-Pierre Jeannerod,Mioara Joldes,Vincent Lefèvre,Guillaume Melquiond,Nathalie Revol,Serge Torres Pdf

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.

Computer Aided Proofs in Analysis

Kenneth R. Meyer,Dieter S. Schmidt
Computer Aided Proofs in Analysis Book PDF

Author : Kenneth R. Meyer,Dieter S. Schmidt
Publisher : Springer Science & Business Media
Page : 264 pages
File Size : 51,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461390923

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Computer Aided Proofs in Analysis by Kenneth R. Meyer,Dieter S. Schmidt Pdf

This IMA Volume in Mathematics and its Applications COMPUTER AIDED PROOFS IN ANALYSIS is based on the proceedings of an IMA Participating Institutions (PI) Conference held at the University of Cincinnati in April 1989. Each year the 19 Participating Institutions select, through a competitive process, several conferences proposals from the PIs, for partial funding. This conference brought together leading figures in a number of fields who were interested in finding exact answers to problems in analysis through computer methods. We thank Kenneth Meyer and Dieter Schmidt for organizing the meeting and editing the proceedings. A vner Friedman Willard Miller, Jr. PREFACE Since the dawn of the computer revolution the vast majority of scientific compu tation has dealt with finding approximate solutions of equations. However, during this time there has been a small cadre seeking precise solutions of equations and rigorous proofs of mathematical results. For example, number theory and combina torics have a long history of computer-assisted proofs; such methods are now well established in these fields. In analysis the use of computers to obtain exact results has been fragmented into several schools.

Computer Arithmetic and Self-Validating Numerical Methods

Christian Ullrich
Computer Arithmetic and Self Validating Numerical Methods Book PDF

Author : Christian Ullrich
Publisher : Academic Press
Page : 316 pages
File Size : 49,8 Mb
Release : 2014-05-10
Category : Computers
ISBN : 9781483267814

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Computer Arithmetic and Self-Validating Numerical Methods by Christian Ullrich Pdf

Notes and Reports in Mathematics in Science and Engineering, Volume VII: Computer Arithmetic and Self-Validating Numerical Methods compiles papers presented at the first international conference on “Computer Arithmetic and Self-Validating Numerical Methods, held in Basel from October 2 to 6, 1989. This book begins by providing a tutorial introduction to computer arithmetic with operations of maximum accuracy, differentiation arithmetic and enclosure methods, and programming languages for self-validating numerical methods. The rest of the chapters discuss the determination of guaranteed bounds for eigenvalues by variational methods and guaranteed inclusion of solutions of differential equations. An appendix covering the IMACS-GAMM resolution on computer arithmetic is provided at the end of this publication. This volume is recommended for researchers and professionals working on computer arithmetic and self-validating numerical methods.

Modern Computer Arithmetic

Richard P. Brent,Paul Zimmermann
Modern Computer Arithmetic Book PDF

Author : Richard P. Brent,Paul Zimmermann
Publisher : Cambridge University Press
Page : 238 pages
File Size : 47,5 Mb
Release : 2010-11-25
Category : Computers
ISBN : 9781139492287

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Modern Computer Arithmetic by Richard P. Brent,Paul Zimmermann Pdf

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.

Proof And Computation Ii: From Proof Theory And Univalent Mathematics To Program Extraction And Verification

Klaus Mainzer,Helmut Schwichtenberg,Peter Michael Schuster
Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification Book PDF

Author : Klaus Mainzer,Helmut Schwichtenberg,Peter Michael Schuster
Publisher : World Scientific
Page : 425 pages
File Size : 46,6 Mb
Release : 2021-07-27
Category : Mathematics
ISBN : 9789811236495

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Proof And Computation Ii: From Proof Theory And Univalent Mathematics To Program Extraction And Verification by Klaus Mainzer,Helmut Schwichtenberg,Peter Michael Schuster Pdf

This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes proof theory, constructive mathematics and type theory, univalent mathematics and point-free approaches to topology, extraction of certified programs from proofs, automated proofs in the automotive industry, as well as the philosophical and historical background of proof theory. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.

Mathematics for Computer Science

Eric Lehman,F. Thomson Leighton,Albert R. Meyer
Mathematics for Computer Science Book PDF

Author : Eric Lehman,F. Thomson Leighton,Albert R. Meyer
Publisher : Unknown
Page : 988 pages
File Size : 41,7 Mb
Release : 2017-03-08
Category : Business & Economics
ISBN : 9888407066

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Mathematics for Computer Science by Eric Lehman,F. Thomson Leighton,Albert R. Meyer Pdf

This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions.

Types for Proofs and Programs

Marino Miculan,Ivan Scagnetto,Furio Honsell
Types for Proofs and Programs Book PDF

Author : Marino Miculan,Ivan Scagnetto,Furio Honsell
Publisher : Springer Science & Business Media
Page : 210 pages
File Size : 48,7 Mb
Release : 2008-05-26
Category : Computers
ISBN : 9783540680840

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Types for Proofs and Programs by Marino Miculan,Ivan Scagnetto,Furio Honsell Pdf

This book constitutes the thoroughly refereed post-conference proceedings of TYPES 2007, the concluding conference of the Types project, held in Cividale del Friuli, Italy, in May 2007. The 13 revised full papers presented were carefully reviewed and selected from 22 submissions. The topic of this last annual workshop of the Types Working Group was formal reasoning and computer programming based on type theory. Great importance was attached to languages and computerized tools for reasoning, and applications in several domains such as analysis of programming languages, certified software, formalization of mathematics and mathematics education.

Software Verification and Formal Methods for ML-Enabled Autonomous Systems

Omri Isac,Radoslav Ivanov,Guy Katz,Nina Narodytska,Laura Nenzi
Software Verification and Formal Methods for ML Enabled Autonomous Systems Book PDF

Author : Omri Isac,Radoslav Ivanov,Guy Katz,Nina Narodytska,Laura Nenzi
Publisher : Springer Nature
Page : 213 pages
File Size : 40,8 Mb
Release : 2022-12-15
Category : Computers
ISBN : 9783031212222

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Software Verification and Formal Methods for ML-Enabled Autonomous Systems by Omri Isac,Radoslav Ivanov,Guy Katz,Nina Narodytska,Laura Nenzi Pdf

This book constitutes the refereed proceedings of the 5th International Workshop on Software Verification and Formal Methods for ML-Enables Autonomous Systems, FoMLAS 2022, and the 15th International Workshop on Numerical Software Verification, NSV 2022, which took place in Haifa, Israel, in July/August 2022. The volume contains 8 full papers from the FoMLAS 2022 workshop and 3 full papers from the NSV 2022 workshop. The FoMLAS workshop is dedicated to the development of novel formal methods techniques to discussing on how formal methods can be used to increase predictability, explainability, and accountability of ML-enabled autonomous systems. NSV 2022 is focusing on the challenges of the verification of cyber-physical systems with machine learning components.

NASA Formal Methods

Jyotirmoy V. Deshmukh,Klaus Havelund,Ivan Perez
NASA Formal Methods Book PDF

Author : Jyotirmoy V. Deshmukh,Klaus Havelund,Ivan Perez
Publisher : Springer Nature
Page : 848 pages
File Size : 55,9 Mb
Release : 2022-05-19
Category : Computers
ISBN : 9783031067730

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NASA Formal Methods by Jyotirmoy V. Deshmukh,Klaus Havelund,Ivan Perez Pdf

This book constitutes the proceedings of the 14th International Symposium on NASA Formal Methods, NFM 2022, held in Pasadena, USA, during May 24-27, 2022. The 33 full and 6 short papers presented in this volume were carefully reviewed and selected from 118submissions. The volume also contains 6 invited papers. The papers deal with advances in formal methods, formal methods techniques, and formal methods in practice. The focus on topics such as interactive and automated theorem proving; SMT and SAT solving; model checking; use of machine learning and probabilistic reasoning in formal methods; formal methods and graphical modeling languages such as SysML or UML; usability of formal method tools and application in industry, etc.