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9783540690801

Fuzzy Implications

by ;
  • ISBN13:

    9783540690801

  • ISBN10:

    3540690808

  • Format: Hardcover
  • Copyright: 2008-11-01
  • Publisher: Springer Verlag
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Summary

"Fuzzy Implications (FIs) generalize the classical implication and play a similar important role in Fuzzy Logic (FL), both in FLn and FLw in the sense of Zadeh. Their importance in applications of FL, viz., Approximate Reasoning (AR), Decision Support Systems, Fuzzy Control (FC), etc., is hard to exaggerate. This treatise is perhaps the first attempt at dealing exclusively with this class of operations." "The book begins with an introduction to FIs along with their desirable properties and investigates their inter-relationships. The remainder of the book is divided into 3 parts. Part I is an analytical study of FIs, detailing the different ways of obtaining FIs, generally called as families of FIs, the various properties they satisfy and the overlaps that exist among them. Part II is an algebraic exploration into the structures that exist on the set of all FIs, while the final part highlights the effecting role FIs with their myriad properties play in the inference schemes in AR." "This book is intended for any researcher in FL operations and it can serve as an auxiliary textbook for different courses on FL or FC. Parts of this book may also be of interest to practitioners in applications of FL, especially in AR or FC, where FIs play a central role."--BOOK JACKET.

Table of Contents

Prefacep. VII
Notations and Some Preliminariesp. XI
An Introduction to Fuzzy Implicationsp. 1
Definition and Basic Examplesp. 1
Continuity of Fuzzy Implicationsp. 7
Basic Properties of Fuzzy Implicationsp. 9
Negations from Fuzzy Implicationsp. 13
Fuzzy Negationsp. 13
Natural Negations of Fuzzy Implicationsp. 17
Laws of Contrapositionp. 20
Reciprocal Fuzzy Implicationsp. 28
Bibliographical Remarksp. 32
Analytical Study of Fuzzy Implications
Fuzzy Implications from Fuzzy Logic Operationsp. 41
Fuzzy Conjunctions: Triangular Normsp. 41
Fuzzy Disjunctions: Triangular Conormsp. 45
Relationships between Negations, T-Norms and T-Conormsp. 49
Natural Negations of T-Norms and T-Conormsp. 49
Laws of Excluded Middle and Contradictionp. 52
De Morgan Triplesp. 55
(S,N)-Implications and S-Implicationsp. 57
Motivation, Definition and Examplesp. 57
Characterizations of (S,N)-Implicationsp. 59
(S,N)-Implications and the Identity Principlep. 64
(S,N)-Implications and the Ordering Propertyp. 64
Intersections between Subfamilies of (S,N)-Implicationsp. 67
R-Implicationsp. 68
Motivation, Definition and Examplesp. 68
Properties of R-Implicationsp. 70
Characterizations and Representations of R-Implicationsp. 75
R-Implications and Laws of Contrapositionp. 85
Intersections between Subfamilies of R-Implicationsp. 89
QL-Implicationsp. 90
Definition, Examples and Basic Propertiesp. 91
QL-Implications and the Exchange Principlep. 99
QL-Implications and the Identity Principlep. 100
QL-Implications and the Ordering Propertyp. 101
QL-Implications and the Law of Contrapositionp. 103
Bibliographical Remarksp. 105
Fuzzy Implications from Generator Functionsp. 109
f-Generated Implicationsp. 109
Definition and Examplesp. 109
Properties of f-Implicationsp. 111
g-Generated Implicationsp. 115
Definition and Examplesp. 116
Properties of g-Implicationsp. 118
Bibliographical Remarksp. 124
Intersections between Families of Fuzzy Implicationsp. 127
Intersections between (S,N)- and R-Implicationsp. 127
Intersections between (S,N)- and QL-Implicationsp. 131
Intersections between R- and QL-Implicationsp. 135
Intersections between Yager's f- and g-Implicationsp. 137
Intersections between Yager's and (S,N)-Implicationsp. 139
Intersections between Yager's and R-Implicationsp. 141
Intersections between Yager's and QL-Implicationsp. 141
Bibliographical Remarksp. 142
Fuzzy Implications from Uninormsp. 145
Uninormsp. 145
Definitions and Examplesp. 145
Pseudo-continuous Uninormsp. 146
Idempotent Uninormsp. 147
Representable Uninormsp. 149
Natural Negations of Fuzzy Implications - Revisitedp. 150
(U,N)-Implicationsp. 151
Definition and Basic Propertiesp. 152
Characterizations of (U,N)-Implicationsp. 155
RU-Implicationsp. 160
Definition and Basic Propertiesp. 160
RU-Implications from Pseudo-continuous Uninormsp. 162
RU-Implications from Representable Uninormsp. 166
RU-Implications from Idempotent Uninormsp. 168
Intersections between (U,N)- and RU-Implicationsp. 172
Intersection between I[subscript U,N] and I[subscript U subscript M]p. 172
Intersection between I[subscript U,N] and I[subscript U subscript R]p. 172
Intersection between I[subscript U,N] and I[subscript U subscript I]p. 173
Intersection between I[subscript U subscript M] and I[subscript U subscript R]p. 174
Intersection between I[subscript U subscript M] and I[subscript U subscript I]p. 174
Intersection between I[subscript U subscript R] and I[subscript U subscript I]p. 175
Bibliographical Remarksp. 175
Algebraic Study of Fuzzy Implications
Algebraic Structures of Fuzzy Implicationsp. 183
Lattice of Fuzzy Implicationsp. 183
Convex Classes of Fuzzy Implicationsp. 188
Conjugacy Classes of Fuzzy Implicationsp. 190
Semigroups of Fuzzy Implicationsp. 197
Composition of Fuzzy Implicationsp. 197
Semigroups of Fuzzy Implicationsp. 201
Bibliographical Remarksp. 205
Fuzzy Implications and Some Functional Equationsp. 207
Contrapositive Symmetrization of Fuzzy Implicationsp. 207
Upper and Lower Contrapositivisationsp. 208
Medium Contrapositivisationp. 212
Distributivity of Fuzzy Implicationsp. 214
On the Equation I(S(x, y), z) = T(I(x, z), I(y, z))p. 215
On the Equation I(T(x, y), z) = S(I(x, z), I(y, z))p. 217
On the Equation I(x, T[subscript 1](y, z)) = T[subscript 2](I(x, y), I(x, z))p. 218
On the Equation I(x, S[subscript 1](y, z)) = S[subscript 2](I(x, y), I(x, z))p. 221
The Law of Importationp. 223
(S,N)-Implications and the Law of Importationp. 224
R-Implications and the Law of Importationp. 225
QL-Implications and the Law of Importationp. 226
f- and g-Implications and the Law of Importationp. 227
Fuzzy Implications and T-Conditionalityp. 229
(S,N)-Implications and T-Conditionalityp. 231
R-Implications and T-Conditionalityp. 232
QL-Implications and T-Conditionalityp. 233
Characterization through Functional Equationsp. 233
Bibliographical Remarksp. 237
Applicational Study of Fuzzy Implications
Fuzzy Implications in Approximate Reasoningp. 245
Approximate Reasoningp. 246
Classical Implication in Inference Schemasp. 246
Fuzzy Implication in Inference Schemasp. 246
Approximate Reasoningp. 247
Fuzzy IF-THEN Rulesp. 247
Possibility Distributionp. 247
Fuzzy Statementsp. 247
Fuzzy IF-THEN Rulesp. 248
Inference Schemes in Approximate Reasoningp. 249
Generalized Modus Ponens (GMP)p. 249
Compositional Rule of Inference (CRI)p. 250
Inference in CRI with Multiple Rulesp. 253
Similarity Based Reasoning (SBR)p. 254
Effectiveness of Inference Schemes in ARp. 257
GMP Rules and ARp. 258
Function Approximation and ARp. 261
Efficiency of Inference Schemes in ARp. 263
Modification of the CRI Inference Algorithmp. 264
Transformation of the Structure of the Rulesp. 268
Bibliographical Remarksp. 275
Appendix
Some Results on Real Functionsp. 281
Referencesp. 289
List of Figuresp. 303
List of Tablesp. 305
Indexp. 307
Table of Contents provided by Ingram. All Rights Reserved.

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