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Design Considerations of Time in Fuzzy Systems Applied Optimization Design Considerations of Time in Fuzzy Systems Applied Optimization Volume 35 Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald Hearn University of Florida, U.S.A. Design Considerations of TilTIe in Fuzzy Systellls by Jemej Virant Faculty of Computer and Information Science. University of Ljubljana. Slovenia .... "Springer-Science+Business Media, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4613-7115-1 ISBN 978-1-4615-4673-3 (eBook) DOI 10.1007/978-1-4615-4673-3 Printed on acid-free paper All Rights Reserved © 2000 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover lst edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. This book is dedicated to understanding the meaning of time in fuzzy structures and systems. Contents List of Figures Xlll List of Tables xxv Foreword xxvii Preface XXIX Acknowledgments xxxv Part I THE FUNDAMENTALS OF FUZZY AND POSSIBILISTIC LOGIC 1. FUZZINESS AND ITS MEASURE 3 1. FUZZY VALUES AND FUZZY VARIABLES 3 2. MEASURES OF FUZZINESS 9 2. FUZZY OPERATIONS AND RELATIONS 19 1. INTRODUCTION 19 2. FUZZY APPROACH TO REASONING 19 3. PARAMETERS, CLASSES AND NORMS 23 4. EXTENDED OPERATIONS 28 5. FUZZY RELATIONS 40 3. FUZZY FUNCTIONS 47 1. INTRODUCTION 47 2. DEFINITION OF FUZZY FUNCTION 47 3. FUZZY OPTIMIZATION 49 3.1 Fuzzy Extremum of Crisp Function 49 3.2 Fuzzy Extremum of Fuzzy Function 53 4. FUZZY INTEGRATION 55 4.1 Integration Over Crisp Interval 55 4.2 Integration Over Fuzzy Interval 58 5. FUZZY DIFFERENTIATION 61 5.1 Differentiation Based on a-notation of Function 61 5.2 Differentiation of Fuzzy Function in LR-notation 63 vii viii DESIGN CONSIDERATIONS OF TIME IN FUZZY SYSTEMS 4. FUZZY ALGORITHMS 65 1. FUZZY RULE LIST 65 2. TRANSFER FUNCTION 68 3. FUZZY MODELING 77 5. PROCESS OF FUZZY INFERENCE 79 1. INTRODUCTION 79 2. FID 79 3. FUZZIFICATION 81 4. FUZZY INFERENCE 83 5. DEFUZZIFICATION 86 5.1 Center-of-Gravity Method 86 5.2 Center-of-Sums Method 88 5.3 First-of-Maxima Method 90 5.4 Height Method 91 5.5 Center-of-Largest-Area Method 91 5.6 Modified COG Method 92 5.7 Schematic Review of the FID Process 93 6. POSSIBILISTIC LOGIC 99 1. INTRODUCTION 99 2. POSSIBILITY AND NECESSITY 101 3. FORMALIZATION OF PROPOSITION 106 4. UNCERTAINTY OF PROPOSITION 108 5. POSSIBILISTIC KNOWLEDGE BASE 114 6. ACCOMPANIED SUBJECTIVITY 120 7. CONCLUSION 121 Part II FUZZY TEMPORAL LOGIC 7. TEMPORALITY OF PROPOSITIONS 125 1. INTRODUCTION 125 2. TIME IN NATURAL LANGUAGE LOGIC 126 3. TIME MODALITIES AND DYNAMICS 130 4. CLASSICAL VIEW OF POSSIBILITY AND NECESSITY IN THE TEMPORAL DOMAIN 132 5. THE MEANING OF TIME IN MODERN SYSTEMS 135 6. THE ROLE OF TIME IN FUZZY INFERENCE 136 7. CATEGORIES OF TEMPORAL OBSERVATIONS 140 8. INCONSISTENCY BASED ON NECESSITY OF FUZZY PROPOSITIONS 141 9. INCONSISTENCY CALCULATION 151 10. INCONSISTENCY BASED ON TEMPORAL NECESSITY- VALUED LOGIC 158 10.1 Fuzzy Temporal Logic 159 Contents lX 10.2 Calculation Based on Fuzzy Formulae 160 11. CONCLUSION 166 8. FUZZINESS OF TIME 167 1. INTRODUCTION 167 2. FUZZY DATE 168 3. BEFORE AND AFTER FUZZY DATE 172 3.1 Non-Relational Observation 172 3.2 Relational Observation 174 3.3 Relation Between Two Dates 185 3.4 Possible Fuzzy Zeros in Observations of Equality 191 4. FUZZY DURATION 194 4.1 Membership Function of Time Interval 195 4.2 Length of fuzzy interval 199 5. EQUALITIES OF FUZZY DATES 208 5.1 K-Equality 208 5.2 6-Equality of dates 215 6. INEQUALITY OF FUZZY DATES 218 7. CALCULATIONS WITH SEVERAL FUZZY DATES 223 8. TRANSPOSED FUZZY SET 227 9. CONCLUSION 229 9. TEMPORALITY IN FUZZY INFERENCE MACHINES 233 1. INTRODUCTION 233 2. TIME DEPENDENT FUZZY SET 234 3. FUZZY TEMPORAL OPERATOR 239 3.1 Basic Definitions 239 3.2 Relation to Given Temporal Knowledge 242 4. INTRODUCING TIME INTO FUZZY RULES 243 5. INTRODUCTION OF TEMPORAL DEPENDENCE INTO INFERENCE MACHINES 245 6. FUZZY MEMORY CELLS 248 6.1 Transition from crisp to fuzzy JK memory cell 249 6.2 Fuzzy T Memory Cell 254 6.2.1 Introduction of FT Memory Cell into Inference Machine 257 6.3 Fuzzy RS Memory Cell 264 6.4 Fuzzy D Memory Cell 267 7. VARIABLES OF FUZZY MEMORY CELLS 269 7.1 Definition of Fuzzy Switching Variable 270 7.2 Input Functions of Fuzzy Memory Cells 275 8. FUZZY AUTOMATA 275 8.1 Fuzzy Relief of Automaton 278 8.2 F JK Memory Cell as Fuzzy Automaton 285 8.3 FT Memory Cell as Fuzzy Automaton 286 9. SOME HARDWARE IMPLEMENTATIONS 292 x DESIGN CONSIDERATIONS OF TIME IN FUZZY SYSTEMS 10.GRAPHS OF FUZZY DATES 295 1. INTRODUCTION 295 2. OPPOSITE AND SYMMETRICAL FUZZY SETS 295 3. NETWORK WITH TEMPORAL CONSTRAINTS 296 4. SETTING THE GRAPH OF THE DATES 298 5. MINIMAL GRAPH OF DATES 303 6. DIAGRAMS WITH FUZZY DELAY 307 6.1 Example of Diagram with Delays 307 6.2 Temporal Possibilities for Fuzzy Set in Diagram with Fuzzy Transition 308 7. FUZZY STATE-TRANSITION GRAPH 316 8. FUZZY PETRI NETS 319 8.1 From Classical to Fuzzy Petri Nets 319 8.2 Fuzzy Time Petri Nets 321 Part III FUZZY SETS COMPUTATION, REPRESENTATION AND SIMULATION l1.CALCULATION AND DISPLAY ERRORS 325 1. CALCULATION OF DEVIATIONS 325 2. ERRORS DUE TO DISCRETE CALCULATIONS 329 3. DISPLAYING MEMBERSHIP FUNCTIONS 333 4. INFLUENCE OF CALCULATION DENSITY 339 5. LINK TO Mathematica ENVIRONMENT 343 6. LINK TO MATLAB ENVIRONMENT 346 12.MATLAB SIMULATIONS OF FUZZY CIRCUITS 351 1. INTRODUCTION 351 2. BUILDING FUZZY MODELS 352 3. LIBRARY OF FUZZY BLOCKS 353 4. OUTLINES OF A SIMULATION 358 5. SIMULATION OF FUZZY MEMORY CELLS 359 5.1 FT Memory Cell 359 5.2 FRS Memory Cell 365 5.3 FD Memory Cell 365 6. SIMULATION OF FUZZY REGISTERS 366 6.1 Clearing the FT Memory Cell 366 6.2 Fuzzy Register FTDCreg 367 6.3 Fuzzy Data Shifting and Delaying 372 7. MODEL OF FUZZY NEURON 377 8. CRISP PROCESSOR - FUZZY PROCESSOR 386 Part IV PROGRAMS Appendices 391 Contents Xl A- Mathematica AND MATLAB PROGRAMS 391 1. APPROACH TO Mathematica SUPPORT 391 2. COMMON PART OF Mathematica PROGRAMS 393 3. FRAME OF Mathematica PROGRAMS 394 4. RUNNING Mathematica PROGRAMS 395 5. FUZZY APPROACH TO MAT LAB SUPPORT 397 6. RUNNING MATLAB PROGRAMS 401 B- LIBRARY OF Mathematica PROGRAMS 407 1. PROGRAM LISTINGS 407 1.1 Program SKUP.MA 407 1.2 Declarations of Basic Vectors 410 1.3 Program ENTRO.MA 410 1.4 Program RELRMA 411 1.5 Program KSUMLMA 412 1.6 Program KINMA.MA 412 1.7 Program SUMIT.MA 413 1.8 Program INMAT.MA 414 1.9 Program REL_SQ.MA 415 1.10 Program INTER.MA 417 1.11 Program DOLIN.MA 418 1.12 Program REAL.MA 420 1.13 Program ENAKMA 421 1.14 Program DENA.MA 423 1.15 Program RAZLLMA 425 1.16 Program ENA.MA 427 1.17 Program DATU.MA 429 1.18 Program TSET.MA 430 1.19 Program HOJKMA 431 1.20 Program IMP.MA 432 1.21 Program HORS.MA 434 1.22 Program HOTOF.MA 436 1.23 Program FAVT .MA 438 1.24 Program HOJKA.MA 440 1.25 Program HOTA.MA 441 1.26 Program HOXY.MA 442 1.27 Program MEA.MA 443 1.28 Program NAPAKMA 448 1.29 Program PRIKMA 449 1.30 Program VEKRV.MA 451 1.31 Program RLPAC.MA 453 C- LIBRARY OF MATLAB PROGRAMS 455 1. PROGRAM LISTINGS 455 1.1 Program MATL.M 455 1.2 Program MFDATA.M 456 1.3 Program XDATA.M 456 1.4 Program FTCELL.MDL 458 References 467 xii DESIGN CONSIDERATIONS OF TIME IN FUZZY SYSTEMS Index 473 List of Figures 1.1 Real line with crisp real number r = 4.5. 3 1.2 Number (4.5) represented as a fuzzy set. 4 1.3 Membership function of fuzzy value {4.5} and terms of fuzzy variable T. 6 1.4 Terms of fuzzy variable V. 6 1.5 Different degrees of fuzziness of fuzzy number (4.5). 7 1.6 FUll support of fuzzy set (4.5). 7 1.7 Various convex shapes of membership functions. 8 1.8 An example of positive fuzzy number A, negative fuzzy number B and fuzzy zero C . 9 1.9 Different degrees of fuzziness of fuzzy sets A and B . 10 1.10 Trapezoidal fuzzy set membership: (top) paramet- ric plot and (bottom) corresponding vector com- panion. 11 1.11 Complement of fuzzy set A for 0 < w S dolz. 13 1.12 Influence of parameter w on fuzzy measure of A. 13 2.1 Venn diagram interpretation of implication. 20 2.2 Presentation of (left) crisp predicate and (right) fuzzy predicate. 24 2.3 Different categories of basic operators. 24 2.4 Yager class of fuzzy complement. 26 2.5 Parameters needed for LR-presentation of a fuzzy set. 30 2.6 Characteristic function L(x) using Eq. 2.36. 34 2.7 Left side of triangle for m = a = 1. 35 2.8 Left side of triangle for a = 2 and m = 1. 35 2.9 Left side of triangle for a = 2 and m = 3. 35 2.10 Right side of triangle for m = f3 = 1.
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