Formal Verification Of Floating Point Hardware Design

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Formal Verification of Floating-Point Hardware Design

Author : David M. Russinoff
Publisher : Springer
ISBN 13 : 9783319955124
Total Pages : 382 pages
Book Rating : 4.9/5 (551 download)

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Book Synopsis Formal Verification of Floating-Point Hardware Design by : David M. Russinoff

Download or read book Formal Verification of Floating-Point Hardware Design written by David M. Russinoff and published by Springer. This book was released on 2018-09-12 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Formal Verification of Floating-Point Hardware Design

Author : David M. Russinoff
Publisher : Springer Nature
ISBN 13 : 3030871819
Total Pages : 448 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Formal Verification of Floating-Point Hardware Design by : David M. Russinoff

Download or read book Formal Verification of Floating-Point Hardware Design written by David M. Russinoff and published by Springer Nature. This book was released on 2022-03-03 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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, Second Edition 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 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, high-level specifications of the basic arithmetic instructions of several major industry-standard floating-point architectures are presented, including all details pertaining to the handling of exceptional conditions. The methodology is illustrated in the comprehensive verification of a variety of state-of-the-art commercial floating-point designs developed by Arm Holdings. This revised edition reflects the evolving microarchitectures and increasing sophistication of Arm processors, and the variation in the design goals of execution speed, hardware area requirements, and power consumption. Many new results have been added to Parts I—III (Register-Transfer Logic, Floating-Point Arithmetic, and Implementation of Elementary Operations), extending the theory and describing new techniques. These were derived as required in the verification of the new RTL designs described in Part V.

Formal Methods for Hardware Verification

Author : Marco Bernardo
Publisher : Springer Science & Business Media
ISBN 13 : 3540343040
Total Pages : 250 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Formal Methods for Hardware Verification by : Marco Bernardo

Download or read book Formal Methods for Hardware Verification written by Marco Bernardo and published by Springer Science & Business Media. This book was released on 2006-05-15 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents 8 papers accompanying the lectures of leading researchers given at the 6th edition of the International School on Formal Methods for the Design of Computer, Communication and Software Systems (SFM 2006). SFM 2006 was devoted to formal techniques for hardware verification and covers several aspects of the hardware design process, including hardware design languages and simulation, property specification formalisms, automatic test pattern generation, symbolic trajectory evaluation, and more.

Handbook of Floating-Point Arithmetic

Author : Jean-Michel Muller
Publisher : Springer Science & Business Media
ISBN 13 : 0817647058
Total Pages : 579 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis Handbook of Floating-Point Arithmetic by : Jean-Michel Muller

Download or read book Handbook of Floating-Point Arithmetic written by Jean-Michel Muller and published by Springer Science & Business Media. This book was released on 2009-11-11 with total page 579 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Formal Verification of Control System Software

Author : Pierre-Loïc Garoche
Publisher : Princeton University Press
ISBN 13 : 0691181306
Total Pages : 230 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis Formal Verification of Control System Software by : Pierre-Loïc Garoche

Download or read book Formal Verification of Control System Software written by Pierre-Loïc Garoche and published by Princeton University Press. This book was released on 2019-05-14 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: An essential introduction to the analysis and verification of control system software The verification of control system software is critical to a host of technologies and industries, from aeronautics and medical technology to the cars we drive. The failure of controller software can cost people their lives. In this authoritative and accessible book, Pierre-Loïc Garoche provides control engineers and computer scientists with an indispensable introduction to the formal techniques for analyzing and verifying this important class of software. Too often, control engineers are unaware of the issues surrounding the verification of software, while computer scientists tend to be unfamiliar with the specificities of controller software. Garoche provides a unified approach that is geared to graduate students in both fields, covering formal verification methods as well as the design and verification of controllers. He presents a wealth of new verification techniques for performing exhaustive analysis of controller software. These include new means to compute nonlinear invariants, the use of convex optimization tools, and methods for dealing with numerical imprecisions such as floating point computations occurring in the analyzed software. As the autonomy of critical systems continues to increase—as evidenced by autonomous cars, drones, and satellites and landers—the numerical functions in these systems are growing ever more advanced. The techniques presented here are essential to support the formal analysis of the controller software being used in these new and emerging technologies.

Formal Verification of Floating-Point Hardware Design

Author : David M. Russinoff
Publisher :
ISBN 13 : 9783030871826
Total Pages : 0 pages
Book Rating : 4.8/5 (718 download)

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Book Synopsis Formal Verification of Floating-Point Hardware Design by : David M. Russinoff

Download or read book Formal Verification of Floating-Point Hardware Design written by David M. Russinoff and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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, Second Edition 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 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, high-level specifications of the basic arithmetic instructions of several major industry-standard floating-point architectures are presented, including all details pertaining to the handling of exceptional conditions. The methodology is illustrated in the comprehensive verification of a variety of state-of-the-art commercial floating-point designs developed by Arm Holdings. This revised edition reflects the evolving microarchitectures and increasing sophistication of Arm processors, and the variation in the design goals of execution speed, hardware area requirements, and power consumption. Many new results have been added to Parts I-III (Register-Transfer Logic, Floating-Point Arithmetic, and Implementation of Elementary Operations), extending the theory and describing new techniques. These were derived as required in the verification of the new RTL designs described in Part V.

Formal Hardware Verification

Author : Thomas Kropf
Publisher : Springer Science & Business Media
ISBN 13 : 9783540634751
Total Pages : 388 pages
Book Rating : 4.6/5 (347 download)

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Book Synopsis Formal Hardware Verification by : Thomas Kropf

Download or read book Formal Hardware Verification written by Thomas Kropf and published by Springer Science & Business Media. This book was released on 1997-08-27 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

A Roadmap for Formal Property Verification

Author : Pallab Dasgupta
Publisher : Springer Science & Business Media
ISBN 13 : 1402047584
Total Pages : 260 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis A Roadmap for Formal Property Verification by : Pallab Dasgupta

Download or read book A Roadmap for Formal Property Verification written by Pallab Dasgupta and published by Springer Science & Business Media. This book was released on 2007-01-19 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Integrating formal property verification (FPV) into an existing design process raises several interesting questions. This book develops the answers to these questions and fits them into a roadmap for formal property verification – a roadmap that shows how to glue FPV technology into the traditional validation flow. The book explores the key issues in this powerful technology through simple examples that mostly require no background on formal methods.

Computer Organization and Design RISC-V Edition

Author : David A. Patterson
Publisher : Morgan Kaufmann
ISBN 13 : 0128122765
Total Pages : 700 pages
Book Rating : 4.1/5 (281 download)

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Book Synopsis Computer Organization and Design RISC-V Edition by : David A. Patterson

Download or read book Computer Organization and Design RISC-V Edition written by David A. Patterson and published by Morgan Kaufmann. This book was released on 2017-05-12 with total page 700 pages. Available in PDF, EPUB and Kindle. Book excerpt: The new RISC-V Edition of Computer Organization and Design features the RISC-V open source instruction set architecture, the first open source architecture designed to be used in modern computing environments such as cloud computing, mobile devices, and other embedded systems. With the post-PC era now upon us, Computer Organization and Design moves forward to explore this generational change with examples, exercises, and material highlighting the emergence of mobile computing and the Cloud. Updated content featuring tablet computers, Cloud infrastructure, and the x86 (cloud computing) and ARM (mobile computing devices) architectures is included. An online companion Web site provides advanced content for further study, appendices, glossary, references, and recommended reading. - Features RISC-V, the first such architecture designed to be used in modern computing environments, such as cloud computing, mobile devices, and other embedded systems - Includes relevant examples, exercises, and material highlighting the emergence of mobile computing and the cloud

Certified Programming with Dependent Types

Author : Adam Chlipala
Publisher : MIT Press
ISBN 13 : 0262317885
Total Pages : 437 pages
Book Rating : 4.2/5 (623 download)

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Book Synopsis Certified Programming with Dependent Types by : Adam Chlipala

Download or read book Certified Programming with Dependent Types written by Adam Chlipala and published by MIT Press. This book was released on 2013-12-06 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: A handbook to the Coq software for writing and checking mathematical proofs, with a practical engineering focus. The technology of mechanized program verification can play a supporting role in many kinds of research projects in computer science, and related tools for formal proof-checking are seeing increasing adoption in mathematics and engineering. This book provides an introduction to the Coq software for writing and checking mathematical proofs. It takes a practical engineering focus throughout, emphasizing techniques that will help users to build, understand, and maintain large Coq developments and minimize the cost of code change over time. Two topics, rarely discussed elsewhere, are covered in detail: effective dependently typed programming (making productive use of a feature at the heart of the Coq system) and construction of domain-specific proof tactics. Almost every subject covered is also relevant to interactive computer theorem proving in general, not just program verification, demonstrated through examples of verified programs applied in many different sorts of formalizations. The book develops a unique automated proof style and applies it throughout; even experienced Coq users may benefit from reading about basic Coq concepts from this novel perspective. The book also offers a library of tactics, or programs that find proofs, designed for use with examples in the book. Readers will acquire the necessary skills to reimplement these tactics in other settings by the end of the book. All of the code appearing in the book is freely available online.

Formal Verification

Author : Erik Seligman
Publisher : Elsevier
ISBN 13 : 0323956122
Total Pages : 426 pages
Book Rating : 4.3/5 (239 download)

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Book Synopsis Formal Verification by : Erik Seligman

Download or read book Formal Verification written by Erik Seligman and published by Elsevier. This book was released on 2023-05-27 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: Formal Verification: An Essential Toolkit for Modern VLSI Design, Second Edition presents practical approaches for design and validation, with hands-on advice to help working engineers integrate these techniques into their work. Formal Verification (FV) enables a designer to directly analyze and mathematically explore the quality or other aspects of a Register Transfer Level (RTL) design without using simulations. This can reduce time spent validating designs and more quickly reach a final design for manufacturing. Building on a basic knowledge of SystemVerilog, this book demystifies FV and presents the practical applications that are bringing it into mainstream design and validation processes. New sections cover advanced techniques, and a new chapter, The Road To Formal Signoff, emphasizes techniques used when replacing simulation work with Formal Verification. After reading this book, readers will be prepared to introduce FV in their organization to effectively deploy FV techniques that increase design and validation productivity.

Computer-Aided Reasoning

Author : Matt Kaufmann
Publisher : Springer
ISBN 13 : 9780792378495
Total Pages : 337 pages
Book Rating : 4.3/5 (784 download)

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Book Synopsis Computer-Aided Reasoning by : Matt Kaufmann

Download or read book Computer-Aided Reasoning written by Matt Kaufmann and published by Springer. This book was released on 2000-06-30 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer-Aided Reasoning: ACL2 Case Studies illustrates how the computer-aided reasoning system ACL2 can be used in productive and innovative ways to design, build, and maintain hardware and software systems. Included here are technical papers written by twenty-one contributors that report on self-contained case studies, some of which are sanitized industrial projects. The papers deal with a wide variety of ideas, including floating-point arithmetic, microprocessor simulation, model checking, symbolic trajectory evaluation, compilation, proof checking, real analysis, and several others. Computer-Aided Reasoning: ACL2 Case Studies is meant for two audiences: those looking for innovative ways to design, build, and maintain hardware and software systems faster and more reliably, and those wishing to learn how to do this. The former audience includes project managers and students in survey-oriented courses. The latter audience includes students and professionals pursuing rigorous approaches to hardware and software engineering or formal methods. Computer-Aided Reasoning: ACL2 Case Studies can be used in graduate and upper-division undergraduate courses on Software Engineering, Formal Methods, Hardware Design, Theory of Computation, Artificial Intelligence, and Automated Reasoning. The book is divided into two parts. Part I begins with a discussion of the effort involved in using ACL2. It also contains a brief introduction to the ACL2 logic and its mechanization, which is intended to give the reader sufficient background to read the case studies. A more thorough, textbook introduction to ACL2 may be found in the companion book, Computer-Aided Reasoning: An Approach. The heart of the book is Part II, where the case studies are presented. The case studies contain exercises whose solutions are on the Web. In addition, the complete ACL2 scripts necessary to formalize the models and prove all the properties discussed are on the Web. For example, when we say that one of the case studies formalizes a floating-point multiplier and proves it correct, we mean that not only can you read an English description of the model and how it was proved correct, but you can obtain the entire formal content of the project and replay the proofs, if you wish, with your copy of ACL2. ACL2 may be obtained from its home page. The results reported in each case study, as ACL2 input scripts, as well as exercise solutions for both books, are available from this page.

Principles of Model Checking

Author : Christel Baier
Publisher : MIT Press
ISBN 13 : 0262304031
Total Pages : 994 pages
Book Rating : 4.2/5 (623 download)

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Book Synopsis Principles of Model Checking by : Christel Baier

Download or read book Principles of Model Checking written by Christel Baier and published by MIT Press. This book was released on 2008-04-25 with total page 994 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive introduction to the foundations of model checking, a fully automated technique for finding flaws in hardware and software; with extensive examples and both practical and theoretical exercises. Our growing dependence on increasingly complex computer and software systems necessitates the development of formalisms, techniques, and tools for assessing functional properties of these systems. One such technique that has emerged in the last twenty years is model checking, which systematically (and automatically) checks whether a model of a given system satisfies a desired property such as deadlock freedom, invariants, and request-response properties. This automated technique for verification and debugging has developed into a mature and widely used approach with many applications. Principles of Model Checking offers a comprehensive introduction to model checking that is not only a text suitable for classroom use but also a valuable reference for researchers and practitioners in the field. The book begins with the basic principles for modeling concurrent and communicating systems, introduces different classes of properties (including safety and liveness), presents the notion of fairness, and provides automata-based algorithms for these properties. It introduces the temporal logics LTL and CTL, compares them, and covers algorithms for verifying these logics, discussing real-time systems as well as systems subject to random phenomena. Separate chapters treat such efficiency-improving techniques as abstraction and symbolic manipulation. The book includes an extensive set of examples (most of which run through several chapters) and a complete set of basic results accompanied by detailed proofs. Each chapter concludes with a summary, bibliographic notes, and an extensive list of exercises of both practical and theoretical nature.

Correct Hardware Design and Verification Methods

Author : Tiziana Margaria
Publisher : Springer
ISBN 13 : 3540447989
Total Pages : 491 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Correct Hardware Design and Verification Methods by : Tiziana Margaria

Download or read book Correct Hardware Design and Verification Methods written by Tiziana Margaria and published by Springer. This book was released on 2003-06-30 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of CHARME 2001, the Eleventh Advanced Research Working Conference on Correct Hardware Design and Veri?cation Methods. CHARME 2001 is the 11th in a series of working conferences devoted to the development and use of leading-edge formal techniques and tools for the design and veri?cation of hardware and hardware-like systems. Previous events in the ‘CHARME’ series were held in Bad Herrenalb (1999), Montreal (1997), Frankfurt (1995), Arles (1993), and Torino (1991). This series of meetings has been organized in cooperation with IFIP WG 10.5 and WG 10.2. Prior meetings, stretching backto the earliest days of formal hardware veri?cation, were held under various names in Miami (1990), Leuven (1989), Glasgow (1988), Grenoble (1986), Edinburgh (1985), and Darmstadt (1984). The convention is now well-established whereby the European CHARME conference alternates with its biennial counterpart, the International Conference on Formal Methods in Computer-Aided Design (FMCAD), which is held on even-numbered years in the USA. The conference tookplace during 4–7 September 2001 at the Institute for System Level Integration in Livingston, Scotland. It was co-hosted by the - stitute and the Department of Computing Science of Glasgow University and co-sponsored by the IFIP TC10/WG10.5 Working Group on Design and En- neering of Electronic Systems. CHARME 2001 also included a scienti?c session and social program held jointly with the 14th International Conference on Th- rem Proving in Higher Order Logics (TPHOLs), which was co-located in nearby Edinburgh.

Program Verification

Author : Timothy T.R. Colburn
Publisher : Springer Science & Business Media
ISBN 13 : 9401117934
Total Pages : 454 pages
Book Rating : 4.4/5 (11 download)

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Book Synopsis Program Verification by : Timothy T.R. Colburn

Download or read book Program Verification written by Timothy T.R. Colburn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt: Among the most important problems confronting computer science is that of developing a paradigm appropriate to the discipline. Proponents of formal methods - such as John McCarthy, C.A.R. Hoare, and Edgar Dijkstra - have advanced the position that computing is a mathematical activity and that computer science should model itself after mathematics. Opponents of formal methods - by contrast, suggest that programming is the activity which is fundamental to computer science and that there are important differences that distinguish it from mathematics, which therefore cannot provide a suitable paradigm. Disagreement over the place of formal methods in computer science has recently arisen in the form of renewed interest in the nature and capacity of program verification as a method for establishing the reliability of software systems. A paper that appeared in Communications of the ACM entitled, `Program Verification: The Very Idea', by James H. Fetzer triggered an extended debate that has been discussed in several journals and that has endured for several years, engaging the interest of computer scientists (both theoretical and applied) and of other thinkers from a wide range of backgrounds who want to understand computer science as a domain of inquiry. The editors of this collection have brought together many of the most interesting and important studies that contribute to answering questions about the nature and the limits of computer science. These include early papers advocating the mathematical paradigm by McCarthy, Naur, R. Floyd, and Hoare (in Part I), others that elaborate the paradigm by Hoare, Meyer, Naur, and Scherlis and Scott (in Part II), challenges, limits and alternatives explored by C. Floyd, Smith, Blum, and Naur (in Part III), and recent work focusing on formal verification by DeMillo, Lipton, and Perlis, Fetzer, Cohn, and Colburn (in Part IV). It provides essential resources for further study. This volume will appeal to scientists, philosophers, and laypersons who want to understand the theoretical foundations of computer science and be appropriately positioned to evaluate the scope and limits of the discipline.

Modern Formal Methods and Applications

Author : Hossam A. Gabbar
Publisher : Springer Science & Business Media
ISBN 13 : 140204223X
Total Pages : 217 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Modern Formal Methods and Applications by : Hossam A. Gabbar

Download or read book Modern Formal Methods and Applications written by Hossam A. Gabbar and published by Springer Science & Business Media. This book was released on 2006-06-28 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: Formal methods are a robust approach for problem solving. It is based on logic and algebraic methods where problems can be formulated in a way that can help to find an appropriate solution. This book shows the basic concepts of formal methods and highlights modern modifications and enhancements to provide a more robust and efficient problem solving tool. Applications are presented from different disciplines such as engineering where the operation of chemical plants is synthesized using formal methods. Computational biology becomes easier and systematic using formal methods. Also, hardware compilation and systems can be managed using formal methods. This book will be helpful for both beginners and experts to get insights and experience on modern formal methods by viewing real applications from different domains.

Correct Hardware Design Methodologies

Author : Paolo Prinetto
Publisher : North Holland
ISBN 13 :
Total Pages : 488 pages
Book Rating : 4.F/5 ( download)

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Book Synopsis Correct Hardware Design Methodologies by : Paolo Prinetto

Download or read book Correct Hardware Design Methodologies written by Paolo Prinetto and published by North Holland. This book was released on 1992 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: The increasing interest in formal verification of hardware is witnessed by academic and industrial research efforts, multinational research projects and conferences. These proceedings contain the papers presented at the Workshop held in Turin, Italy from 12-14 June, 1991. Papers focus on the practical impact of formal techniques in VLSI design and verification, semantics for hardware description languages, proof environments and methodologies. Preliminary results of the ESPRIT Basic Research Action 3216 CHARME are discussed. Finite State Machines and their verification, semantics for hardware description languages, proof environments and methodologies, as well as examples of verified circuits and Design for Verifiability are also presented.