Fuzzy Implications

by Baczynski, Michal; Jayaram, Balasubramaniam

• ISBN13: 9783540690801
• ISBN10: 3540690808
• Format: Hardcover
• Copyright: 2008-11-01
• Publisher: Springer Verlag

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

Preface	p. VII
Notations and Some Preliminaries	p. XI
An Introduction to Fuzzy Implications	p. 1
Definition and Basic Examples	p. 1
Continuity of Fuzzy Implications	p. 7
Basic Properties of Fuzzy Implications	p. 9
Negations from Fuzzy Implications	p. 13
Fuzzy Negations	p. 13
Natural Negations of Fuzzy Implications	p. 17
Laws of Contraposition	p. 20
Reciprocal Fuzzy Implications	p. 28
Bibliographical Remarks	p. 32
Analytical Study of Fuzzy Implications
Fuzzy Implications from Fuzzy Logic Operations	p. 41
Fuzzy Conjunctions: Triangular Norms	p. 41
Fuzzy Disjunctions: Triangular Conorms	p. 45
Relationships between Negations, T-Norms and T-Conorms	p. 49
Natural Negations of T-Norms and T-Conorms	p. 49
Laws of Excluded Middle and Contradiction	p. 52
De Morgan Triples	p. 55
(S,N)-Implications and S-Implications	p. 57
Motivation, Definition and Examples	p. 57
Characterizations of (S,N)-Implications	p. 59
(S,N)-Implications and the Identity Principle	p. 64
(S,N)-Implications and the Ordering Property	p. 64
Intersections between Subfamilies of (S,N)-Implications	p. 67
R-Implications	p. 68
Motivation, Definition and Examples	p. 68
Properties of R-Implications	p. 70
Characterizations and Representations of R-Implications	p. 75
R-Implications and Laws of Contraposition	p. 85
Intersections between Subfamilies of R-Implications	p. 89
QL-Implications	p. 90
Definition, Examples and Basic Properties	p. 91
QL-Implications and the Exchange Principle	p. 99
QL-Implications and the Identity Principle	p. 100
QL-Implications and the Ordering Property	p. 101
QL-Implications and the Law of Contraposition	p. 103
Bibliographical Remarks	p. 105
Fuzzy Implications from Generator Functions	p. 109
f-Generated Implications	p. 109
Definition and Examples	p. 109
Properties of f-Implications	p. 111
g-Generated Implications	p. 115
Definition and Examples	p. 116
Properties of g-Implications	p. 118
Bibliographical Remarks	p. 124
Intersections between Families of Fuzzy Implications	p. 127
Intersections between (S,N)- and R-Implications	p. 127
Intersections between (S,N)- and QL-Implications	p. 131
Intersections between R- and QL-Implications	p. 135
Intersections between Yager's f- and g-Implications	p. 137
Intersections between Yager's and (S,N)-Implications	p. 139
Intersections between Yager's and R-Implications	p. 141
Intersections between Yager's and QL-Implications	p. 141
Bibliographical Remarks	p. 142
Fuzzy Implications from Uninorms	p. 145
Uninorms	p. 145
Definitions and Examples	p. 145
Pseudo-continuous Uninorms	p. 146
Idempotent Uninorms	p. 147
Representable Uninorms	p. 149
Natural Negations of Fuzzy Implications - Revisited	p. 150
(U,N)-Implications	p. 151
Definition and Basic Properties	p. 152
Characterizations of (U,N)-Implications	p. 155
RU-Implications	p. 160
Definition and Basic Properties	p. 160
RU-Implications from Pseudo-continuous Uninorms	p. 162
RU-Implications from Representable Uninorms	p. 166
RU-Implications from Idempotent Uninorms	p. 168
Intersections between (U,N)- and RU-Implications	p. 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 Remarks	p. 175
Algebraic Study of Fuzzy Implications
Algebraic Structures of Fuzzy Implications	p. 183
Lattice of Fuzzy Implications	p. 183
Convex Classes of Fuzzy Implications	p. 188
Conjugacy Classes of Fuzzy Implications	p. 190
Semigroups of Fuzzy Implications	p. 197
Composition of Fuzzy Implications	p. 197
Semigroups of Fuzzy Implications	p. 201
Bibliographical Remarks	p. 205
Fuzzy Implications and Some Functional Equations	p. 207
Contrapositive Symmetrization of Fuzzy Implications	p. 207
Upper and Lower Contrapositivisations	p. 208
Medium Contrapositivisation	p. 212
Distributivity of Fuzzy Implications	p. 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 Importation	p. 223
(S,N)-Implications and the Law of Importation	p. 224
R-Implications and the Law of Importation	p. 225
QL-Implications and the Law of Importation	p. 226
f- and g-Implications and the Law of Importation	p. 227
Fuzzy Implications and T-Conditionality	p. 229
(S,N)-Implications and T-Conditionality	p. 231
R-Implications and T-Conditionality	p. 232
QL-Implications and T-Conditionality	p. 233
Characterization through Functional Equations	p. 233
Bibliographical Remarks	p. 237
Applicational Study of Fuzzy Implications
Fuzzy Implications in Approximate Reasoning	p. 245
Approximate Reasoning	p. 246
Classical Implication in Inference Schemas	p. 246
Fuzzy Implication in Inference Schemas	p. 246
Approximate Reasoning	p. 247
Fuzzy IF-THEN Rules	p. 247
Possibility Distribution	p. 247
Fuzzy Statements	p. 247
Fuzzy IF-THEN Rules	p. 248
Inference Schemes in Approximate Reasoning	p. 249
Generalized Modus Ponens (GMP)	p. 249
Compositional Rule of Inference (CRI)	p. 250
Inference in CRI with Multiple Rules	p. 253
Similarity Based Reasoning (SBR)	p. 254
Effectiveness of Inference Schemes in AR	p. 257
GMP Rules and AR	p. 258
Function Approximation and AR	p. 261
Efficiency of Inference Schemes in AR	p. 263
Modification of the CRI Inference Algorithm	p. 264
Transformation of the Structure of the Rules	p. 268
Bibliographical Remarks	p. 275
Appendix
Some Results on Real Functions	p. 281
References	p. 289
List of Figures	p. 303
List of Tables	p. 305
Index	p. 307
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