Triangular Norms
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Triangular Norms
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Author : Erich Peter Klement
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Triangular Norms written by Erich Peter Klement and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Philosophy categories.
The history of triangular norms started with the paper "Statistical metrics" [Menger 1942]. The main idea of Karl Menger was to construct metric spaces where probability distributions rather than numbers are used in order to de scribe the distance between two elements of the space in question. Triangular norms (t-norms for short) naturally came into the picture in the course of the generalization of the classical triangle inequality to this more general set ting. The original set of axioms for t-norms was considerably weaker, including among others also the functions which are known today as triangular conorms. Consequently, the first field where t-norms played a major role was the theory of probabilistic metric spaces ( as statistical metric spaces were called after 1964). Berthold Schweizer and Abe Sklar in [Schweizer & Sklar 1958, 1960, 1961] provided the axioms oft-norms, as they are used today, and a redefinition of statistical metric spaces given in [Serstnev 1962]led to a rapid development of the field. Many results concerning t-norms were obtained in the course of this development, most of which are summarized in the monograph [Schweizer & Sklar 1983]. Mathematically speaking, the theory of (continuous) t-norms has two rather independent roots, namely, the field of (specific) functional equations and the theory of (special topological) semigroups.
Associative Functions Triangular Norms And Copulas
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Author : Claudi Alsina
language : en
Publisher: World Scientific
Release Date : 2006-02-22
Associative Functions Triangular Norms And Copulas written by Claudi Alsina and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-22 with Mathematics categories.
The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the “triangular norms” introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis.This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.
Neutrosophic Triangular Norms And Their Derived Residuated Lattices
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Author : Qingqing Hu
language : en
Publisher: Infinite Study
Release Date :
Neutrosophic Triangular Norms And Their Derived Residuated Lattices written by Qingqing Hu and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
Neutrosophic triangular norms (t-norms) and their residuated lattices are not only the main research object of neutrosophic set theory, but also the core content of neutrosophic logic. Neutrosophic implications are important operators of neutrosophic logic. Neutrosophic residual implications based on neutrosophic t-norms can be applied to the fields of neutrosophic inference and neutrosophic control. In this paper, neutrosophic t-norms, neutrosophic residual implications, and the residuated lattices derived from neutrosophic t-norms are investigated deeply. First of all, the lattice and its corresponding system are proved to be a complete lattice and a De Morgan algebra, respectively. Second, the notions of neutrosophic t-norms are introduced on the complete lattice discussed earlier. The basic concepts and typical examples of representable and non-representable neutrosophic t-norms are obtained. Naturally, De Morgan neutrosophic triples are defined for the duality of neutrosophic t-norms and neutrosophic t-conorms with respect to neutrosophic negators. Third, neutrosophic residual implications generated from neutrosophic t-norms and their basic properties are investigated. Furthermore, residual neutrosophic t-norms are proved to be infinitely -distributive, and then some important properties possessed by neutrosophic residual implications are given. Finally, a method for producing neutrosophic t-norms from neutrosophic implications is presented, and the residuated lattices are constructed on the basis of neutrosophic t-norms and neutrosophic residual implications.
Triangular Norm Based Measures And Games With Fuzzy Coalitions
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Author : D. Butnariu
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Triangular Norm Based Measures And Games With Fuzzy Coalitions written by D. Butnariu and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Business & Economics categories.
This book aims to present, in a unified approach, a series of mathematical results con cerning triangular norm-based measures and a class of cooperative games with Juzzy coalitions. Our approach intends to emphasize that triangular norm-based measures are powerful tools in exploring the coalitional behaviour in 'such games. They not and simplify some technical aspects of the already classical axiomatic the only unify ory of Aumann-Shapley values, but also provide new perspectives and insights into these results. Moreover, this machinery allows us to obtain, in the game theoretical context, new and heuristically meaningful information, which has a significant impact on balancedness and equilibria analysis in a cooperative environment. From a formal point of view, triangular norm-based measures are valuations on subsets of a unit cube [0, 1]X which preserve dual binary operations induced by trian gular norms on the unit interval [0, 1]. Triangular norms (and their dual conorms) are algebraic operations on [0,1] which were suggested by MENGER [1942] and which proved to be useful in the theory of probabilistic metric spaces (see also [WALD 1943]). The idea of a triangular norm-based measure was implicitly used under various names: vector integrals [DVORETZKY, WALD & WOLFOWITZ 1951], prob abilities oj Juzzy events [ZADEH 1968], and measures on ideal sets [AUMANN & SHAPLEY 1974, p. 152].
Logical Algebraic Analytic And Probabilistic Aspects Of Triangular Norms
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Author : Erich Peter Klement
language : en
Publisher: Elsevier
Release Date : 2005-03-25
Logical Algebraic Analytic And Probabilistic Aspects Of Triangular Norms written by Erich Peter Klement and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-03-25 with Mathematics categories.
This volume gives a state of the art of triangular norms which can be used for the generalization of several mathematical concepts, such as conjunction, metric, measure, etc. 16 chapters written by leading experts provide a state of the art overview of theory and applications of triangular norms and related operators in fuzzy logic, measure theory, probability theory, and probabilistic metric spaces.Key Features:- Complete state of the art of the importance of triangular norms in various mathematical fields- 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications- Chapter authors are leading authorities in their fields- Triangular norms on different domains (including discrete, partially ordered) are described- Not only triangular norms but also related operators (aggregation operators, copulas) are covered- Book contains many enlightening illustrations· Complete state of the art of the importance of triangular norms in various mathematical fields· 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications· Chapter authors are leading authorities in their fields· Triangular norms on different domains (including discrete, partially ordered) are described· Not only triangular norms but also related operators (aggregation operators, copulas) are covered· Book contains many enlightening illustrations
Fuzzy If Then Rules In Computational Intelligence
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Author : Da Ruan
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Fuzzy If Then Rules In Computational Intelligence written by Da Ruan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
During the last three decades, interest has increased significantly in the representation and manipulation of imprecision and uncertainty. Perhaps the most important technique in this area concerns fuzzy logic or the logic of fuzziness initiated by L. A. Zadeh in 1965. Since then, fuzzy logic has been incorporated into many areas of fundamental science and into the applied sciences. More importantly, it has been successful in the areas of expert systems and fuzzy control. The main body of this book consists of so-called IF-THEN rules, on which experts express their knowledge with respect to a certain domain of expertise. Fuzzy IF-THEN Rules in Computational Intelligence: Theory and Applications brings together contributions from leading global specialists who work in the domain of representation and processing of IF-THEN rules. This work gives special attention to fuzzy IF-THEN rules as they are being applied in computational intelligence. Included are theoretical developments and applications related to IF-THEN problems of propositional calculus, fuzzy predicate calculus, implementations of the generalized Modus Ponens, approximate reasoning, data mining and data transformation, techniques for complexity reduction, fuzzy linguistic modeling, large-scale application of fuzzy control, intelligent robotic control, and numerous other systems and practical applications. This book is an essential resource for engineers, mathematicians, and computer scientists working in fuzzy sets, soft computing, and of course, computational intelligence.
Computational Intelligence
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Author : Leszek Rutkowski
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-05-24
Computational Intelligence written by Leszek Rutkowski and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-05-24 with Technology & Engineering categories.
This quite simply superb book focuses on various techniques of computational intelligence, both single ones and those which form hybrid methods. These techniques are today commonly applied to issues of artificial intelligence. The book presents methods of knowledge representation using different techniques, namely the rough sets, type-1 fuzzy sets and type-2 fuzzy sets. Next up, various neural network architectures are presented and their learning algorithms are derived. Then, the family of evolutionary algorithms is discussed, including connections between these techniques and neural networks and fuzzy systems. Finally, various methods of data partitioning and algorithms of automatic data clustering are given and new neuro-fuzzy architectures are studied and compared.
Handbook Of Granular Computing
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Author : Witold Pedrycz
language : en
Publisher: John Wiley & Sons
Release Date : 2008-07-31
Handbook Of Granular Computing written by Witold Pedrycz and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-07-31 with Technology & Engineering categories.
Although the notion is a relatively recent one, the notions and principles of Granular Computing (GrC) have appeared in a different guise in many related fields including granularity in Artificial Intelligence, interval computing, cluster analysis, quotient space theory and many others. Recent years have witnessed a renewed and expanding interest in the topic as it begins to play a key role in bioinformatics, e-commerce, machine learning, security, data mining and wireless mobile computing when it comes to the issues of effectiveness, robustness and uncertainty. The Handbook of Granular Computing offers a comprehensive reference source for the granular computing community, edited by and with contributions from leading experts in the field. Includes chapters covering the foundations of granular computing, interval analysis and fuzzy set theory; hybrid methods and models of granular computing; and applications and case studies. Divided into 5 sections: Preliminaries, Fundamentals, Methodology and Algorithms, Development of Hybrid Models and Applications and Case Studies. Presents the flow of ideas in a systematic, well-organized manner, starting with the concepts and motivation and proceeding to detailed design that materializes in specific algorithms, applications and case studies. Provides the reader with a self-contained reference that includes all pre-requisite knowledge, augmented with step-by-step explanations of more advanced concepts. The Handbook of Granular Computing represents a significant and valuable contribution to the literature and will appeal to a broad audience including researchers, students and practitioners in the fields of Computational Intelligence, pattern recognition, fuzzy sets and neural networks, system modelling, operations research and bioinformatics.
Fuzzy Logic Theory And Applications Part I And Part Ii
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Author : Lotfi A Zadeh
language : en
Publisher: World Scientific Publishing
Release Date : 2018-12-04
Fuzzy Logic Theory And Applications Part I And Part Ii written by Lotfi A Zadeh and has been published by World Scientific Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-12-04 with Computers categories.
Nowadays, voluminous textbooks and monographs in fuzzy logic are devoted only to separate or some combination of separate facets of fuzzy logic. There is a lack of a single book that presents a comprehensive and self-contained theory of fuzzy logic and its applications.Written by world renowned authors, Lofti Zadeh, also known as the Father of Fuzzy Logic, and Rafik Aliev, who are pioneers in fuzzy logic and fuzzy sets, this unique compendium includes all the principal facets of fuzzy logic such as logical, fuzzy-set-theoretic, epistemic and relational. Theoretical problems are prominently illustrated and illuminated by numerous carefully worked-out and thought-through examples.This invaluable volume will be a useful reference guide for academics, practitioners, graduates and undergraduates in fuzzy logic and its applications.
Advances In Data Analysis With Computational Intelligence Methods
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Author : Adam E Gawęda
language : en
Publisher: Springer
Release Date : 2017-09-21
Advances In Data Analysis With Computational Intelligence Methods written by Adam E Gawęda and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-21 with Technology & Engineering categories.
This book is a tribute to Professor Jacek Żurada, who is best known for his contributions to computational intelligence and knowledge-based neurocomputing. It is dedicated to Professor Jacek Żurada, Full Professor at the Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, J.B. Speed School of Engineering, University of Louisville, Kentucky, USA, as a token of appreciation for his scientific and scholarly achievements, and for his longstanding service to many communities, notably the computational intelligence community, in particular neural networks, machine learning, data analyses and data mining, but also the fuzzy logic and evolutionary computation communities, to name but a few. At the same time, the book recognizes and honors Professor Żurada’s dedication and service to many scientific, scholarly and professional societies, especially the IEEE (Institute of Electrical and Electronics Engineers), the world’s largest professional technical professional organization dedicated to advancing science and technology in a broad spectrum of areas and fields. The volume is divided into five major parts, the first of which addresses theoretic, algorithmic and implementation problems related to the intelligent use of data in the sense of how to derive practically useful information and knowledge from data. In turn, Part 2 is devoted to various aspects of neural networks and connectionist systems. Part 3 deals with essential tools and techniques for intelligent technologies in systems modeling and Part 4 focuses on intelligent technologies in decision-making, optimization and control, while Part 5 explores the applications of intelligent technologies.