Zadeh / Fu / Tanaka Fuzzy Sets and Their Applications to Cognitive and Decision Processes

Proceedings of the U.S.-Japan Seminar on Fuzzy Sets and Their Applications, Held at the University of California, Berkeley, California, July 1-4, 1974
1. Auflage 2014
ISBN: 978-1-4832-6591-9
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 506 Seiten

Fuzzy Sets and Their Applications to Cognitive and Decision Processes contains the proceedings of the U.S.-Japan Seminar on Fuzzy Sets and Their Applications, held at the University of California in Berkeley, California, on July 1-4, 1974. The seminar provided a forum for discussing a broad spectrum of topics related to the theory of fuzzy sets, ranging from its mathematical aspects to applications in human cognition, communication, decision making, and engineering systems analysis. Comprised of 19 chapters, this book begins with an introduction to the calculus of fuzzy restrictions, followed by a discussion on fuzzy programs and their execution. Subsequent chapters focus on fuzzy relations, fuzzy graphs, and their applications to clustering analysis; risk and decision making in a fuzzy environment; fractionally fuzzy grammars and their application to pattern recognition; and applications of fuzzy sets in psychology. An approach to pattern recognition and associative memories using fuzzy logic is also described. This monograph will be of interest to students and practitioners in the fields of computer science, engineering, psychology, and applied mathematics.

Autoren/Hrsg.

Zadeh, Lotfi A.

Fu, King-Sun

Tanaka, Kokichi

Inhaltsverzeichnis

1;Front Cover;1
2;Fuzzy Sets and Their Applications to Cognitive and Decision Processes;4
3;Copyright Page;5
4;Table of Contents;6
5;CONTRIBUTORS;8
6;PREFACE;10
7;CHAPTER 1. CALCULUS OF FUZZY RESTRICTIONS;12
7.1;ABSTRACT;12
7.2;1. INTRODUCTION;13
7.3;2. CALCULUS OF FUZZY RESTRICTIONS;17
7.4;3. APPROXIMATE REASONING (AR);29
7.5;4. CONCLUDING REMARKS;36
7.6;REFERENCES;36
7.7;APPENDIX;38
8;CHAPTER 2. FUZZY PROGRAMS AND THEIR EXECUTION;52
8.1;1. INTRODUCTION;52
8.2;2. GENERALIZED FUZZY MACHINES;53
8.3;3. EXECUTION PROCEDURE OF FUZZY PROGRAMS;58
8.4;4. SIMULATION OF HUMAN DRIVER'S BEHAVIOR;62
8.5;5. SIMULATION OF CHARACTER GENERATION;72
8.6;6. CONCLUSION;85
8.7;REFERENCES;86
9;CHAPTER 3. FUZZY GRAPHS;88
9.1;ABSTRACT;88
9.2;1. INTRODUCTION;88
9.3;2. FUZZY RELATIONS ON FUZZY SETS;89
9.4;3. COMPOSITION OF FUZZY RELATIONS;90
9.5;4. REFLEXIVITY AND SYMMETRY;93
9.6;5. TRANSITIVITY;94
9.7;6. FUZZY GRAPHS;96
9.8;7. PATHS AND CONNECTEDNESS;97
9.9;8. CLUSTERS;99
9.10;9. BRIDGES AND CUTNODES;101
9.11;10. FORESTS AND TREES;102
9.12;REFERENCES;106
10;CHAPTER 4. FUZZINESS IN INFORMATIVE LOGICS;108
10.1;1. INTRODUCTION;108
10.2;2. INFORMATIVE LOGICS;110
10.3;3. MECHANICAL INTELLIGENCE AND GENETIC EPISTEMOLOGY;115
10.4;4. VAGUENESS AND FUZZINESS IN INFORMATIVE LOGICS;118
10.5;5. A NEW DIRECTION FOR A GENERALIZATION OF FUZZINESS CONCEPT TO BE USED IN DEVELOPING INFORMATION SCIENCE APPROACHES;125
10.6;REFERENCES;131
11;CHAPTER 5. FUZZY RELATIONS, FUZZY GRAPHS, AND THEIR APPLICATIONS TO CLUSTERING ANALYSIS;136
11.1;I. INTRODUCTION;136
11.2;II. PRELIMINARIES;136
11.3;III. AN ALGEBRA OF FUZZY RELATIONS;138
11.4;IV. FUZZY GRAPHS;143
11.5;V. SYMMETRIC FUZZY GRAPHS;148
11.6;VI. APPLICATION TO CLUSTERING ANALYSIS;154
11.7;REFERENCES;159
12;CHAPTER 6. CONDITIONAL FUZZY MEASURES AND THEIR APPLICATIONS;162
12.1;1. INTRODUCTION;162
12.2;2. FUZZY MEASURES AND INTEGRALS;164
12.3;3. TRANSITION OF FUZZY PHENOMENA;169
12.4;4. APPLICATIONS;174
12.5;5. CONCLUSION;180
12.6;ACKNOWLEDGEMENTS;180
12.7;REFERENCES;180
13;CHAPTER 7. FUZZY TOPOLOGY;182
13.1;ABSTRACT;182
13.2;I. INTRODUCTION;182
13.3;II. BASIC DEFINITIONS AND PROPERTIES;183
13.4;III. COMPACTNESS AND COUNTABILITY;186
13.5;IV. PRODUCT AND QUOTIENT SPACES;187
13.6;V. LOCAL PROPERTIES;191
13.7;VI. NORMALITY AND UNIFORMITY;196
13.8;REFERENCES;201
14;CHAPTER 8. INTERPRETATION AND EXECUTION OF FUZZY PROGRAMS;202
14.1;ABSTRACT;202
14.2;1. INTRODUCTION;202
14.3;2. FUZZY SETS AND FUZZY RELATIONS;203
14.4;3. OPERATIONS ON FUZZY SETS;209
14.5;4. FUZZY PROGRAMS;213
14.6;5. EXECUTIONS (INTERPRETATIONS) OF FUZZY PROGRAMS;214
14.7;6. MODELING ILL-DEFINED PROCEDURES BY FUZZY PROGRAMS;227
14.8;7. CONCLUDING REMARKS;228
14.9;REFERENCES;229
15;CHAPTER 9. ON RISK AND DECISION MAKING IN A FUZZY ENVIRONMENT;230
15.1;ABSTRACT;230
15.2;INTRODUCTION;230
15.3;CONVEXITY AND FUZZY MAPPING;231
15.4;THEOREMS ON CONVEXITY;232
15.5;DEFINITION OF BINARY OPERATION;234
15.6;MULTISTAGE DECISION PROCESS;234
15.7;RISK AND OPTIMUM DECISION;235
15.8;APPLICATIONS;236
15.9;ACKNOWLEDGEMENT;237
15.10;REFERENCES;237
16;CHAPTER 10. AN AXIOMATIC APPROACH TO RATIONAL DECISION MAKING IN A FUZZY ENVIRONMENT;238
16.1;ABSTRACT;238
16.2;I. INTRODUCTION;238
16.3;II. BASIC ASSUMPTIONS;240
16.4;III. RATIONAL AGGREGATES;251
16.5;IV. ALTERNATIVE ASSUMPTIONS;261
16.6;V. DISCUSSIONS;264
16.7;REFERENCES;265
17;CHAPTER 11. DECISION-MAKING AND ITS GOAL IN A FUZZY ENVIRONMENT;268
17.1;1. INTRODUCTION;268
17.2;2. PSEUDO SIMILARITY RELATIONS;270
17.3;3. O-DECISION PROBLEMS;273
17.4;4. 1-DECISION PROBLEMS;277
17.5;5. N-DECISION PROBLEMS;282
17.6;6. CONCLUDING REMARKS;285
17.7;REFERENCES;286
17.8;APPENDIX;286
18;CHAPTER 12. RECOGNITION OF FUZZY LANGUAGES;290
18.1;ABSTRACT;290
18.2;1. INTRODUCTION;290
18.3;2. FUZZY LANGUAGES;292
18.4;3. CUT-POINTS AND THEIR REPRESENTATION;292
18.5;4. F-RECOGNITIONS BY MACHINES;294
18.6;5. ISOLATED CUT-POINTS;299
18.7;6. A FUZZY LANGUAGE WHICH IS NOT F-RECOGNIZED BY A MACHINE IN DT2;306
18.8;7. RECURSIVE FUZZY LANGUAGES;308
18.9;ACKNOWLEDGMENT;310
18.10;REFERENCES;310
19;CHAPTER 13. ON THE DESCRIPTION OF FUZZY MEANING OF CONTEXT-FREE LANGUAGE;312
19.1;1. INTRODUCTION;312
19.2;2. TREES AND PSEUDOTERMS;313
19.3;3. FUZZY DENDROLANGUAGE GENERATING SYSTEMS;315
19.4;4. NORMAL FORM OF F-CFDS;317
19.5;5. CHARACTERIZATION OF SETS OF DERIVATION TREES OF FUZZY CONTEXT-FREE GRAMMARS;321
19.6;6. FUZZY TREE AUTOMATON;326
19.7;7. FUZZY TREE TRANSDUCER;330
19.8;8. FUZZY MEANING OF CONTEXT-FREE LANGUAGE;334
19.9;ACKNOWLEDGEMENT;339
19.10;REFERENCES;339
20;CHAPTER 14. FRACTIONALLY FUZZY GRAMMARS WITH APPLICATION TO PATTERN RECOGNITION;340
20.1;ABSTRACT;340
20.2;I. INTRODUCTION;340
20.3;II. BACKGROUND AND NOTATION;342
20.4;III. FRACTIONALLY FUZZY GRAMMARS;346
20.5;IV. A PATTERN RECOGNITION EXPERIMENT;354
20.6;V. CONCLUSION;360
20.7;REFERENCES;360
21;CHAPTER 15. TOWARD INTEGRATED COGNITIVE SYSTEMS,
WHICH MUST MAKE FUZZY DECISIONS
ABOUT FUZZY PROBLEMS;364
21.1;INTRODUCTION;364
21.2;HIGHER-LEVEL FUZZY PROBLEMS;384
21.3;THE STEP-BY-STEP DEVELOPMENT OF MODELS OF INTELLIGENT MIND/BRAINS;387
21.4;SUMMARY AND CONCLUSIONS;389
21.5;APPENDIX A: THE SEER-2 PROGRAM;391
21.6;APPENDIX B: CHARACTERIZING TRANSFORMS THAT FORM SEER's MEMORY NETWORK;395
21.7;APPENDIX C: A NOTE ON EASEy PROGRAMS;400
21.8;REFERENCES;401
22;CHAPTER 16. APPLICATIONS OF FUZZY SETS IN PSYCHOLOGY;406
22.1;1. INTRODUCTION;406
22.2;2. BACKGROUND: SUMMARY OF PREVIOUS RESULTS ON THE USE OF FUZZY SET THEORY IN PSYCHOLOGY;407
22.3;3. A CONCEPTUAL ISSUE;410
22.4;4. AN EXPERIMENT;413
22.5;5. RESULTS;414
22.6;6. CONCLUSIONS;418
22.7;REFERENCES;418
23;CHAPTER 17. EXPERIMENTAL APPROACH TO FUZZY SIMULATION OF MEMORIZING, FORGETTING AND INFERENCE PROCESS;420
23.1;1. INTRODUCTION;420
23.2;2. PROPOSITION OF A WHOLE MODEL OF HUMAN DECISION-MAKING PROCESS;421
23.3;3. FUZZY FORMULATION OF DECISION-MAKING PROCESS;423
23.4;4. FUZZY SIMULATION OF MEMORIZING- AND FORGETTING PROCESSES;425
23.5;5. FUZZY SIMULATION OF INFERENCE PROCESS BASED ON MEMORY;430
23.6;6. CONCLUSIONS;437
23.7;REFERENCES;438
24;CHAPTER 18. ON FUZZY ROBOT PLANNING;440
24.1;INTRODUCTION;440
24.2;1. ROBUSTNESS;442
24.3;2. NATURAL LANGUAGE UNDERSTANDING;443
24.4;3. SYSTEM OVERVIEW AND OBJECTIVES;447
24.5;4. SOME DETAILS;450
24.6;5. CONCLUSIONS AND EXTENSIONS;455
24.7;REFERENCES;457
25;CHAPTER 19. AN APPROACH TO PATTERN RECOGNITION
AND ASSOCIATIVE MEMORIES
USING FUZZY LOGIC;460
25.1;ABSTRACT;460
25.2;1. INTRODUCTION;460
25.3;2. MULTICATEGORY PATTERN CLASSIFICATION;463
25.4;3. ASSOCIATIVE MEMORIES I;470
25.5;4. ASSOCIATIVE MEMORIES II;473
25.6;5. COMPUTER SIMULATION;476
25.7;6. CONCLUDING REMARKS;485
25.8;REFERENCES;487
26;BIBLIOGRAPHY ON FUZZY SETS AND THEIR APPLICATIONS;488

Leseproben

CALCULUS OF FUZZY RESTRICTIONS
L.A. Zadeh*,     Department of Electrical Engineering and Computer Sciences, University of California Berkeley, California 94720 ABSTRACT
A fuzzy restriction may be visualized as an elastic constraint on the values that may be assigned to a variable. In terms of such restrictions, the meaning of a proposition of the form “x is P,” where x is the name of an object and P is a fuzzy set, may be expressed as a relational assignment equation of the form R(A(x)) = P, where A(x) is an implied attribute of x, R is a fuzzy restriction on x, and P is the unary fuzzy relation which is assigned to R. For example, “Stella is young,” where young is a fuzzy subset of the real line, translates into R(Age(Stella))= young.The calculus of fuzzy restrictions is concerned, in the main, with (a) translation of propositions of various types into relational assignment equations, and (b) the study of transformations of fuzzy restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations. An important application of the calculus of fuzzy restrictions relates to what might be called approximate reasoning, that is, a type of reasoning which is neither very exact nor very inexact. The main ideas behind this application are outlined and illustrated by examples. 1 INTRODUCTION
During the past decade, the theory of fuzzy sets has developed in a variety of directions, finding applications in such diverse fields as taxonomy, topology, linguistics, automata theory, logic, control theory, game theory, information theory, psychology, pattern recognition, medicine, law, decision analysis, system theory and information retrieval. A common thread that runs through most of the applications of the theory of fuzzy sets relates to the concept of a fuzzy restriction - that is, a fuzzy relation which acts as an elastic constraint on the values that may be assigned to a variable. Such restrictions appear to play an important role in human cognition, especially in situations involving concept formation, pattern recognition, and decision-making in fuzzy or uncertain environments. As its name implies, the calculus of fuzzy restrictions is essentially a body of concepts and techniques for dealing with fuzzy restrictions in a systematic fashion. As such, it may be viewed as a branch of the theory of fuzzy relations, in which it plays a role somewhat analogous to that of the calculus of probabilities in probability theory. However, a more specific aim of the calculus of fuzzy restrictions is to furnish a conceptual basis for fuzzy logic and what might be called approximate reasoning [1], that is, a type of reasoning which is neither very exact nor very inexact. Such reasoning plays a basic role in human decision-making because it provides a way of dealing with problems which are too complex for precise solution. However, approximate reasoning is more than a method of last recourse for coping with insurmountable complexities. It is, also, a way of simplifying the performance of tasks in which a high degree of precision is neither needed nor required. Such tasks pervade much of what we do on both conscious and subconscious levels. What is a fuzzy restriction? To illustrate its meaning in an informal fashion, consider the following proposition (in which italicized words represent fuzzy concepts): young_ (1.1) (1.1) gray hair_ (1.2) (1.2) approximately equal_  in  height. (1.3) (1.3) Starting with (1.1), let Age (Tosi) denote a numerically-valued variable which ranges over the interval [0,100]. With this interval regarded as our universe of discourse U, young may be interpreted as the label of a fuzzy subset1 of U which is characterized by a compatibility function, µyoung’ of the form shown in Fig. 1.1. Thus, the degree to which a numerical age, say u = 28, is compatible with the concept of young is 0.7, while the compatibilities of 30 and 35 with young are 0.5 and 0.2, respectively. (The age at which the compatibility takes the value 0.5 is the crossover point of young.) Equivalently, the function µyoung may be viewed as the membership function of the fuzzy set young, with the value of µyoung at u representing the grade of membership of u in young.
Figure 1.1 Compatibility Function of young. Since young is a fuzzy set with no sharply defined boundaries, the conventional interpretation of the proposition “Tosi is young,” namely, “Tosi is a member of the class of young men,” is not meaningful if membership in a set is interpreted in its usual mathematical sense. To circumvent this difficulty, we shall view (1.1)as an assertion of a restriction on the possible values of Tosi’s age rather than as an assertion concerning the membership of Tosi in a class of individuals. Thus, on denoting the restriction on the age of Tosi by R(Age(Tosi)), (1.1)may be expressed as an assignment equation young_ (1.4) (1.4) in which the fuzzy set young (or, equivalently, the unary fuzzy relation young) is assigned to the restriction on the variable Age (Tosi). In this instance, the restriction R(Age(Tosi)) is a fuzzy restriction by virtue of the fuzziness of the set young. Using the same point of view, (1.2)may be expressed as  gray_ (1.5) (1.5) Thus, in this case, the fuzzy set gray is assigned as a value to the fuzzy restriction on the variable Color (Hair(Ted)). In the case of (1.1)and (1.2), the fuzzy restriction has the form of a fuzzy set or, equivalently, a unary fuzzy relation. In the case of (1.3), we have two variables to consider, namely, Height (Sakti) and Height (Kapali). Thus, in this instance, the assignment equation takes the form  approximately_ equal_ (1.6) (1.6) in which approximately equal is a binary fuzzy relation characterized by a compatibility matrix µapproximately equal (u, v) such as shown in Table 1.2. Table 1.2 Compatibility matrix of the fuzzy Relation approximately equal. 5'6 1 0.8 0.6 0.2 0 0 5'8 0.8 1 0.9 0.7 0.3 0 5'10 0.6 0.9 1 0.9 0.7 0 6 0.2 0.7 0.9 1 0.9 0.8 6'2 0 0.3 0.7 0.9 1 0.9 6'4 0 0 0 0.8 0.9 1 Thus, if Sakti’s height is 5'8 and Kapali’s is 5'10, then the degree to which they are approximately equal is 0.9. The restrictions involved in (1.1), (1.2)and (1.3)are unrelated in the sense that the restriction on the age of Tosi has no bearing on the color of Ted’s hair or the height of Sakti and Kapali. More generally, however, the restrictions may be interrelated, as in the following example. small_ (1.7) (1.7) approximately equal_ (1.8) (1.8) In terms of the fuzzy restrictions on u and v, (1.7)and (1.8)translate into the assignment equations small_ (1.9) (1.9) approximately equal_ (1.10) (1.10) where R (u) and R (u, v) denote the restrictions on u and (u, v), respectively. As will be shown in Section 2, from the knowledge of a fuzzy restriction on u and a fuzzy restriction on (u, v) we can deduce a fuzzy restriction on v. Thus, in the case of (1.9)and (1.10), we can assert that °R(u,v) = small_°approximately equal_ (1.11) (1.11) where denotes the composition2 of fuzzy relations. The rule by which (1.11)is inferred from (1.9)and (1.10)is called the...