Uncertain Rule-based Fuzzy Logic Systems

Introduction and New Directions

Author: Jerry M. Mendel

Publisher: Prentice Hall

ISBN: 9780130409690

Category: Computers

Page: 555

View: 6858

Preface Uncertainty is the fabric that makes life interesting. For millenia human beings have developed strategies to cope with a plethora of uncertainties, never absolutely sure what the consequences would be, but hopeful that the deleterious effects of those uncertainties could be minimized. This book presents a complete methodology for accomplishing this within the framework of fuzzy logic (FL). This is not the original FL, but is an expanded and richer FL, one that contains the original FL within it. The original FL, founded by Lotfi Zadeh, has been around for more than 35 years, as of the year 2000, and yet it is unable to handle uncertainties. Byhandle, I meanto model and minimize the effect of. That the original FL—type-1 FL—cannot do this sounds paradoxical because the word fuzzy has the connotation of uncertainty. The expanded FL—type-2 FL—is able to handle uncertainties because it can model them and minimize their effects. And, if all uncertainties disappear, type-2 FL reduces to type-1 FL, in much the same way that if randomness disappears, probability reduces to determinism. Although many applications were found for type-1 FL, it is its application torule-based systemsthat has most significantly demonstrated its importance as a powerful design methodology. Such rule-based fuzzy logic systems (FLSs), both type-1 and type-2, are what this book is about. In it I show how to use FL in new ways and how to effectively solve problems that are awash in uncertainties. FL has already been applied in numerous fields, in many of which uncertainties are present (e.g., signal processing, digital communications, computer and communication networks, diagnostic medicine, operations research, financial investing, control, etc.). Hence, the results in this book can immediately be used in all of these fields. To demonstrate the performance advantages for type-2 FLSs over their type-1 counterparts, when uncertainties are present, I describe and provide results for the following applications in this book: forecasting of time series, knowledge-mining using surveys, classification of video data working directly with compressed data, equalization of time-varying nonlinear digital communication channels, overcoming co-channel interference and intersymbol interference for time-varying nonlinear digital communication channels, and connection admission control for asynchronous transfer mode networks. No control applications have been included, because to date type-2 FL has not yet been applied to them; hence, this book is not about FL control, although its methodologies may someday be applicable to it. I have organized this book into four parts. Part 1—Preliminaries— contains four chapters that provide background materials about uncertainty, membership functions, and two case studies (forecasting of time-series and knowledge mining using surveys) that are carried throughout the book. Part 2—Type-1 Fuzzy Logic Systems—contains two chapters that are included to provide the underlying basis for the new type-2 FLSs, so that we can compare type-2 results for our case studies with type-1 results. Part 3—Type-2 Fuzzy Sets—contains three chapters, each of which focuses on a different aspect of such sets. Part 4—Type-2 Fuzzy Logic Systems—which is the heart of the book, contains five chapters, four having to do with different architectures for a FLS and how to handle different kinds of uncertainties within them, and one having to do primarily with four specific applications of type-2 FLSs. This book can be read by anyone who has an undergraduate BS degree and should be of great interest to computer scientists and engineers who already use or want to use rule-based systems and are concerned with how to handle uncertainties about such systems. I have included many worked-out examples in the text, and have also included homework problems at the end of most chapters so that the book can be used in a classroom setting as well as a technical reference. Here are some specific ways that this book can be used: For the person totally unfamiliar with FL who wants a quick introduction to it, read the Supplement to Chapter 1 and Chapter 5 (Sections 5.1-5.8). For the person who wants an in-depth treatment of type-1 rule-based FLSs, read the Supplement to Chapter 1 and Chapters 4-6. For the person who is only interested in type-2 fuzzy set theory, read Chapters 3, 7-9, and Appendices A and B. For a person who wants to give a course on rule-based fuzzy logic systems, use Chapters 1-12 and 13 (if time permits). Chapter 14 should be of interest to people with a background in digital communications, pattern recognition, or communication networks and will suggest projects for a course. For a person who is a proponent of Takagi-Sugeno-Kang (TSK) fuzzy systems and wants to see what their type-2 counterparts look like, read Chapters 3, 7-9, and 13. For a person who is interested in forecasting of time-series and wants to get a quick overview of the benefits to modeling uncertainties on forecasting performance when using rule-based forecasters, read Chapters 4 (Section 4.2), 5 (Section 5.10), 6 (Section 6.7), 10 (Section 10.11), 11 (Section 11.5), and 12 (Section 12.5). For a person who is interested in knowledge mining and wants to get a quick overview of the benefits to modeling uncertainties on judgment making when using rule-based advisors, read Chapters 4 (Section 4.3), 5 (Section 5.11), and 10 (Section 10.12). So that people will start using type-2 FL as soon as possible, I have made free software available online for implementing and designing type-1 and type-2 FLSs. It is MATLAB-based (MATLAB is a registered trademark of The MathWorks, Inc.), was developed by my former PhD students Nilesh Karnik and Qilian Liang, and can be reached at:http://sipi.usc.edu/. A computation section, which directs the reader to very specific M-files, appears at the end of most chapters of this book. Appendix C summarizes all of the M-files so that the reader can see the forest from the trees.

Uncertain Rule-Based Fuzzy Systems

Introduction and New Directions, 2nd Edition

Author: Jerry M. Mendel

Publisher: Springer

ISBN: 3319513702

Category: Technology & Engineering

Page: 684

View: 9573

The second edition of this textbook provides a fully updated approach to fuzzy sets and systems that can model uncertainty — i.e., “type-2” fuzzy sets and systems. The author demonstrates how to overcome the limitations of classical fuzzy sets and systems, enabling a wide range of applications from time-series forecasting to knowledge mining to control. In this new edition, a bottom-up approach is presented that begins by introducing classical (type-1) fuzzy sets and systems, and then explains how they can be modified to handle uncertainty. The author covers fuzzy rule-based systems – from type-1 to interval type-2 to general type-2 – in one volume. For hands-on experience, the book provides information on accessing MatLab and Java software to complement the content. The book features a full suite of classroom material.

Advances in Type-2 Fuzzy Sets and Systems

Theory and Applications

Author: Alireza Sadeghian,Jerry Mendel,Hooman Tahayori

Publisher: Springer

ISBN: 1461466660

Category: Computers

Page: 262

View: 3400

This book explores recent developments in the theoretical foundations and novel applications of general and interval type-2 fuzzy sets and systems, including: algebraic properties of type-2 fuzzy sets, geometric-based definition of type-2 fuzzy set operators, generalizations of the continuous KM algorithm, adaptiveness and novelty of interval type-2 fuzzy logic controllers, relations between conceptual spaces and type-2 fuzzy sets, type-2 fuzzy logic systems versus perceptual computers; modeling human perception of real world concepts with type-2 fuzzy sets, different methods for generating membership functions of interval and general type-2 fuzzy sets, and applications of interval type-2 fuzzy sets to control, machine tooling, image processing and diet. The applications demonstrate the appropriateness of using type-2 fuzzy sets and systems in real world problems that are characterized by different degrees of uncertainty.

Computational Intelligence Paradigms

Theory & Applications using MATLAB

Author: S. Sumathi,Surekha Paneerselvam

Publisher: CRC Press

ISBN: 9781439809037

Category: Computers

Page: 851

View: 7767

Offering a wide range of programming examples implemented in MATLAB®, Computational Intelligence Paradigms: Theory and Applications Using MATLAB® presents theoretical concepts and a general framework for computational intelligence (CI) approaches, including artificial neural networks, fuzzy systems, evolutionary computation, genetic algorithms and programming, and swarm intelligence. It covers numerous intelligent computing methodologies and algorithms used in CI research. The book first focuses on neural networks, including common artificial neural networks; neural networks based on data classification, data association, and data conceptualization; and real-world applications of neural networks. It then discusses fuzzy sets, fuzzy rules, applications of fuzzy systems, and different types of fused neuro-fuzzy systems, before providing MATLAB illustrations of ANFIS, classification and regression trees, fuzzy c-means clustering algorithms, fuzzy ART map, and Takagi–Sugeno inference systems. The authors also describe the history, advantages, and disadvantages of evolutionary computation and include solved MATLAB programs to illustrate the implementation of evolutionary computation in various problems. After exploring the operators and parameters of genetic algorithms, they cover the steps and MATLAB routines of genetic programming. The final chapter introduces swarm intelligence and its applications, particle swarm optimization, and ant colony optimization. Full of worked examples and end-of-chapter questions, this comprehensive book explains how to use MATLAB to implement CI techniques for the solution of biological problems. It will help readers with their work on evolution dynamics, self-organization, natural and artificial morphogenesis, emergent collective behaviors, swarm intelligence, evolutionary strategies, genetic programming, and the evolution of social behaviors.

Intelligent Systems for Information Processing: From Representation to Applications

Author: B. Bouchon-Meunier,L. Foulloy,Ronald R. Yager

Publisher: Elsevier

ISBN: 9780080525655

Category: Mathematics

Page: 488

View: 2257

Intelligent systems are required to enhance the capacities being made available to us by the internet and other computer based technologies. The theory necessary to help providing solutions to difficult problems in the construction of intelligent systems are discussed. In particular, attention is paid to situations in which the available information and data may be imprecise, uncertain, incomplete or of a linguistic nature. Various methodologies to manage such information are discussed. Among these are the probabilistic, possibilistic, fuzzy, logical, evidential and network-based frameworks. One purpose of the book is not to consider these methodologies separately, but rather to consider how they can be used cooperatively to better represent the multiplicity of modes of information. Topics in the book include representation of imperfect knowledge, fundamental issues in uncertainty, reasoning, information retrieval, learning and mining, as well as various applications. Key Features: • Tools for construction of intelligent systems • Contributions by world leading experts • Fundamental issues and applications • New technologies for web searching • Methods for modeling uncertain information • Future directions in web technologies • Transversal to methods and domains

Flexible Neuro-Fuzzy Systems

Structures, Learning and Performance Evaluation

Author: Leszek Rutkowski

Publisher: Springer Science & Business Media

ISBN: 1402080433

Category: Computers

Page: 279

View: 1251

Flexible Neuro-Fuzzy Systems is the first professional literature about the new class of powerful, flexible fuzzy systems. The author incorporates various flexibility parameters to the construction of neuro-fuzzy systems. This approach dramatically improves their performance, allowing the systems to perfectly represent the pattern encoded in data. Flexible Neuro-Fuzzy Systems is the only book that proposes a flexible approach to fuzzy modeling and fills the gap in existing literature. This book introduces new fuzzy systems which outperform previous approaches to system modeling and classification, and has the following features: -Provides a framework for unification, construction and development of neuro-fuzzy systems; -Presents complete algorithms in a systematic and structured fashion, facilitating understanding and implementation, -Covers not only advanced topics but also fundamentals of fuzzy sets, -Includes problems and exercises following each chapter, -Illustrates the results on a wide variety of simulations, -Provides tools for possible applications in business and economics, medicine and bioengineering, automatic control, robotics and civil engineering.

Advances in Neural Networks – ISNN 2015

12th International Symposium on Neural Networks, ISNN 2015, Jeju, South Korea, October 15-18, 2015, Proceedings

Author: Xiaolin Hu,Yousheng Xia,Yunong Zhang,Dongbin Zhao

Publisher: Springer

ISBN: 331925393X

Category: Computers

Page: 510

View: 5709

The volume LNCS 9377 constitutes the refereed proceedings of the 12th International Symposium on Neural Networks, ISNN 2015, held in Jeju, South Korea in October 2015. The 55 revised full papers presented were carefully reviewed and selected from 97 submissions. These papers cover many topics of neural network-related research including intelligent control, neurodynamic analysis, memristive neurodynamics, computer vision, signal processing, machine learning, and optimization.

Information Processing and Management of Uncertainty

15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2014, Montpellier, France, July 15-19, 2014. Proceedings

Author: Anne Laurent,Olivier Strauss,Bernadette Bouchon-Meunier,Ronald R. Yager

Publisher: Springer

ISBN: 3319087959

Category: Computers

Page: 603

View: 9621

These three volumes (CCIS 442, 443, 444) constitute the proceedings of the 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2014, held in Montpellier, France, July 15-19, 2014. The 180 revised full papers presented together with five invited talks were carefully reviewed and selected from numerous submissions. The papers are organized in topical sections on uncertainty and imprecision on the web of data; decision support and uncertainty management in agri-environment; fuzzy implications; clustering; fuzzy measures and integrals; non-classical logics; data analysis; real-world applications; aggregation; probabilistic networks; recommendation systems and social networks; fuzzy systems; fuzzy logic in boolean framework; management of uncertainty in social networks; from different to same, from imitation to analogy; soft computing and sensory analysis; database systems; fuzzy set theory; measurement and sensory information; aggregation; formal methods for vagueness and uncertainty in a many-valued realm; graduality; preferences; uncertainty management in machine learning; philosophy and history of soft computing; soft computing and sensory analysis; similarity analysis; fuzzy logic, formal concept analysis and rough set; intelligent databases and information systems; theory of evidence; aggregation functions; big data - the role of fuzzy methods; imprecise probabilities: from foundations to applications; multinomial logistic regression on Markov chains for crop rotation modelling; intelligent measurement and control for nonlinear systems.

Fuzzy Hybrid Computing in Construction Engineering and Management

Theory and Applications

Author: Aminah Robinson Fayek

Publisher: Emerald Group Publishing

ISBN: 1787439968

Category: Technology & Engineering

Page: 468

View: 5407

This book is a guide for students, researchers, and practitioners to the latest developments in fuzzy hybrid computing in construction engineering and management. It discusses basic theory related to fuzzy logic and fuzzy hybrid computing, their application in a range of practical construction problems, and emerging and future research trends.

Trends in Applied Intelligent Systems

23rd International Conference on Industrial Engineering and Other Applications of Applied Intelligent Systems, IEA/AIE 2010, Cordoba, Spain, June 1-4, 2010, Proceedings

Author: Nicolás García-Pedrajas,Francisco Herrera,Colin Fyfe,José Manuel Benítez Sánchez,Moonis Ali

Publisher: Springer

ISBN: 364213033X

Category: Computers

Page: 674

View: 2713


Perceptual Computing

Aiding People in Making Subjective Judgments

Author: Jerry Mendel,Dongrui Wu

Publisher: John Wiley & Sons

ISBN: 9780470599631

Category: Computers

Page: 256

View: 2551

Explains for the first time how "computing with words" can aid in making subjective judgments Lotfi Zadeh, the father of fuzzy logic, coined the phrase "computing with words" (CWW) to describe a methodology in which the objects of computation are words and propositions drawn from a natural language. Perceptual Computing explains how to implement CWW to aid in the important area of making subjective judgments, using a methodology that leads to an interactive device—a "Perceptual Computer"—that propagates random and linguistic uncertainties into the subjective judgment in a way that can be modeled and observed by the judgment maker. This book focuses on the three components of a Perceptual Computer—encoder, CWW engines, and decoder—and then provides detailed applications for each. It uses interval type-2 fuzzy sets (IT2 FSs) and fuzzy logic as the mathematical vehicle for perceptual computing, because such fuzzy sets can model first-order linguistic uncertainties whereas the usual kind of fuzzy sets cannot. Drawing upon the work on subjective judgments that Jerry Mendel and his students completed over the past decade, Perceptual Computing shows readers how to: Map word-data with its inherent uncertainties into an IT2 FS that captures these uncertainties Use uncertainty measures to quantify linguistic uncertainties Compare IT2 FSs by using similarity and rank Compute the subsethood of one IT2 FS in another such set Aggregate disparate data, ranging from numbers to uniformly weighted intervals to nonuniformly weighted intervals to words Aggregate multiple-fired IF-THEN rules so that the integrity of word IT2 FS models is preserved Free MATLAB-based software is also available online so readers can apply the methodology of perceptual computing immediately, and even try to improve upon it. Perceptual Computing is an important go-to for researchers and students in the fields of artificial intelligence and fuzzy logic, as well as for operations researchers, decision makers, psychologists, computer scientists, and computational intelligence experts.

Principles of Bioseparations Engineering

Author: Raja Ghosh

Publisher: World Scientific Publishing Company

ISBN: 981310676X

Category: Technology & Engineering

Page: 284

View: 7087

Bioseparations engineering deals with the scientific and engineering principles involved in large-scale separation and purification of biological products. It is a key component of most chemical engineering/biotechnology/bioprocess engineering programmes. This book discusses the underlying principles of bioseparations engineering written from the perspective of an undergraduate course. It covers membrane based bioseparations in much more detail than some of the other books on bioseparations engineering. Based largely on the lecture notes the author developed to teach the course, this book is especially suitable for use as an undergraduate level textbook, as most other textbooks are targeted at graduate students.

Handbook On Computational Intelligence (In 2 Volumes)

Author: Angelov Plamen Parvanov

Publisher: World Scientific

ISBN: 9814675024

Category: Computers

Page: 964

View: 7184

With the Internet, the proliferation of Big Data, and autonomous systems, mankind has entered into an era of 'digital obesity'. In this century, computational intelligence, such as thinking machines, have been brought forth to process complex human problems in a wide scope of areas — from social sciences, economics and biology, medicine and social networks, to cyber security.The Handbook of Computational Intelligence (in two volumes) prompts readers to look at these problems from a non-traditional angle. It takes a step by step approach, supported by case studies, to explore the issues that have arisen in the process. The Handbook covers many classic paradigms, as well as recent achievements and future promising developments to solve some of these very complex problems. Volume one explores the subjects of fuzzy logic and systems, artificial neural networks, and learning systems. Volume two delves into evolutionary computation, hybrid systems, as well as the applications of computational intelligence in decision making, the process industry, robotics, and autonomous systems.This work is a 'one-stop-shop' for beginners, as well as an inspirational source for more advanced researchers. It is a useful resource for lecturers and learners alike.

Uncertainty Modeling in Vibration, Control and Fuzzy Analysis of Structural Systems

Author: Bilal M. Ayyub,Ard‚shir Guran,Achintya Haldar

Publisher: World Scientific

ISBN: 9810231342

Category: Technology & Engineering

Page: 356

View: 1186

This book gives an overview of the current state of uncertainty modeling in vibration, control, and fuzzy analysis of structural and mechanical systems. It is a coherent compendium written by leading experts and offers the reader a sampling of exciting research areas in several fast-growing branches in this field. Uncertainty modeling and analysis are becoming an integral part of system definition and modeling in many fields. The book consists of ten chapters that report the work of researchers, scientists and engineers on theoretical developments and diversified applications in engineering systems. They deal with modeling for vibration, control, and fuzzy analysis of structural and mechanical systems under uncertain conditions. The book designed for readers who are familiar with the fundamentals and wish to study a particular topic or use the book as an authoritative reference. It gives readers a sophisticated toolbox for tackling modeling problems in mechanical and structural systems in real-world situations. The book is part of a series on Stability, Vibration and Control of Structures, and provides vital information in these areas.

Intelligent Systems for Engineers and Scientists, Third Edition

Author: Adrian A. Hopgood

Publisher: CRC Press

ISBN: 1439865965

Category: Computers

Page: 451

View: 1982

The third edition of this bestseller examines the principles of artificial intelligence and their application to engineering and science, as well as techniques for developing intelligent systems to solve practical problems. Covering the full spectrum of intelligent systems techniques, it incorporates knowledge-based systems, computational intelligence, and their hybrids. Using clear and concise language, Intelligent Systems for Engineers and Scientists, Third Edition features updates and improvements throughout all chapters. It includes expanded and separated chapters on genetic algorithms and single-candidate optimization techniques, while the chapter on neural networks now covers spiking networks and a range of recurrent networks. The book also provides extended coverage of fuzzy logic, including type-2 and fuzzy control systems. Example programs using rules and uncertainty are presented in an industry-standard format, so that you can run them yourself. The first part of the book describes key techniques of artificial intelligence—including rule-based systems, Bayesian updating, certainty theory, fuzzy logic (types 1 and 2), frames, objects, agents, symbolic learning, case-based reasoning, genetic algorithms, optimization algorithms, neural networks, hybrids, and the Lisp and Prolog languages. The second part describes a wide range of practical applications in interpretation and diagnosis, design and selection, planning, and control. The author provides sufficient detail to help you develop your own intelligent systems for real applications. Whether you are building intelligent systems or you simply want to know more about them, this book provides you with detailed and up-to-date guidance. Check out the significantly expanded set of free web-based resources that support the book at: http://www.adrianhopgood.com/aitoolkit/

A New Paradigm of Knowledge Engineering by Soft Computing

Author: Liya Ding

Publisher: World Scientific

ISBN: 9789812794604

Category: Computers

Page: 388

View: 7061

Soft computing (SC) consists of several computing paradigms, including neural networks, fuzzy set theory, approximate reasoning, and derivative-free optimization methods such as genetic algorithms. The integration of those constituent methodologies forms the core of SC. In addition, the synergy allows SC to incorporate human knowledge effectively, deal with imprecision and uncertainty, and learn to adapt to unknown or changing environments for better performance. Together with other modern technologies, SC and its applications exert unprecedented influence on intelligent systems that mimic human intelligence in thinking, learning, reasoning, and many other aspects. Knowledge engineering (KE), which deals with knowledge acquisition, representation, validation, inferencing, explanation, and maintenance, has made significant progress recently, owing to the indefatigable efforts of researchers. Undoubtedly, the hot topics of data mining and knowledge/data discovery have injected new life into the classical AI world. This book tells readers how KE has been influenced and extended by SC and how SC will be helpful in pushing the frontier of KE further. It is intended for researchers and graduate students to use as a reference in the study of knowledge engineering and intelligent systems. The reader is expected to have a basic knowledge of fuzzy logic, neural networks, genetic algorithms, and knowledge-based systems. Contents: Knowledge Engineering and Soft Computing OCo An Introduction (L-Y Ding); Fuzzy Knowledge-Based Systems: Linguistic Integrity: A Framework for Fuzzy Modeling OCo AFRELI Algorithm (J Espinosa & J Vandewalle); A New Approach to Acquisition of Comprehensible Fuzzy Rules (H Ohno & T Furuhashi); Fuzzy Rule Generation with Fuzzy Singleton-Type Reasoning Method (Y Shi & M Mizumoto); Antecedent Validity Adaptation Principle for Table Look-Up Scheme (P-T Chan & A B Rad); Fuzzy Spline Interpolation in Sparse Fuzzy Rule Bases (M F Kawaguchi & M Miyakoshi); Revision Principle Applied for Approximate Reasoning (L-Y Ding et al.); Handling Null Queries with Compound Fuzzy Attributes (S-L Wang & Y-J Tsai); Fuzzy System Description Language (K Otsuka et al.); Knowledge Representation, Integration, and Discovery by Soft Computing: Knowledge Representation and Similarity Measure in Learning a Vague Legal Concept (M Q Xu et al.); Trend Fuzzy Sets and Recurrent Fuzzy Rules for Ordered Dataset Modelling (J F Baldwin et al.); Approaches to the Design of Classification Systems from Numerical Data and Linguistic Knowledge (H Ishibuchi et al.); A Clustering Based on Self-Organizing Map and Knowledge Discovery by Neural Network (K Nakagawa et al.); Probabilistic Rough Induction (J-Z Dong et al.); Data Mining via Linguistic Summaries of Databases: An Interactive Approach (J Kacprzyk & S Zadrozny); and other papers. Readership: Graduate students, researchers and lecturers in knowledge engineering and soft computing."

Fuzzy Cognitive Maps for Applied Sciences and Engineering

From Fundamentals to Extensions and Learning Algorithms

Author: Elpiniki I. Papageorgiou

Publisher: Springer Science & Business Media

ISBN: 3642397395

Category: Computers

Page: 395

View: 7810

Fuzzy Cognitive Maps (FCM) constitute cognitive models in the form of fuzzy directed graphs consisting of two basic elements: the nodes, which basically correspond to “concepts” bearing different states of activation depending on the knowledge they represent, and the “edges” denoting the causal effects that each source node exercises on the receiving concept expressed through weights. Weights take values in the interval [-1,1], which denotes the positive, negative or neutral causal relationship between two concepts. An FCM can be typically obtained through linguistic terms, inherent to fuzzy systems, but with a structure similar to the neural networks, which facilitates data processing, and has capabilities for training and adaptation. During the last 10 years, an exponential growth of published papers in FCMs was followed showing great impact potential. Different FCM structures and learning schemes have been developed, while numerous studies report their use in many contexts with highly successful modeling results. The aim of this book is to fill the existing gap in the literature concerning fundamentals, models, extensions and learning algorithms for FCMs in knowledge engineering. It comprehensively covers the state-of-the-art FCM modeling and learning methods, with algorithms, codes and software tools, and provides a set of applications that demonstrate their various usages in applied sciences and engineering.


The Inventor of Algebra

Author: Corona Brezina

Publisher: Rosen Central

ISBN: 9781435837485

Category: Biography & Autobiography

Page: 114

View: 9486

Al-Khwarizmi is arguably the most important mathematician of the Middle Ages. He developed two distinct branches of mathematics, both of which owe their name to him: algebra and algorithms. This carefully crafted biography shines a long-overdue light on these achievements, documents Khwarizmi's contributions to geography and astronomy, and paints a picture of life in the ninth-century Muslim Empire. Supports history-social science context standards mandating exploration of intellectual exchanges and contributions of Muslim scholars, and their influence on the science, geography, mathematics, philosophy, and medicine of later civilizations.

Rule-based expert systems

the MYCIN experiments of the Stanford Heuristic Programming Project

Author: Bruce G. Buchanan,Edward Hance Shortliffe

Publisher: Addison Wesley Publishing Company


Category: Computers

Page: 748

View: 7297