Introduction to Fuzzy Logic

Learn more about the history, foundations, and applications of fuzzy logic in this comprehensive resource by an academic leader

Introduction to Fuzzy Logic delivers a high-level but accessible introduction to the rapidly growing and evolving field of fuzzy logic and its applications. Distinguished engineer, academic, and author James K. Peckol covers a wide variety of practical topics, including the differences between crisp and fuzzy logic, the people and professions who find fuzzy logic useful, and the advantages of using fuzzy logic.

While the book assumes a solid foundation in embedded systems, including basic logic design, and C/C++ programming, it is written in a practical and easy-to-read style that engages the reader and assists in learning and retention. The author includes introductions of threshold and perceptron logic to further enhance the applicability of the material contained within.

After introducing readers to the topic with a brief description of the history and development of the field, Introduction to Fuzzy Logic goes on to discuss a wide variety of foundational and advanced topics, like:

• A review of Boolean algebra, including logic minimization with algebraic means and Karnaugh maps
• A discussion of crisp sets, including classic set membership, set theory and operations, and basic classical crisp set properties
• A discussion of fuzzy sets, including the foundations of fuzzy sets logic, set membership functions, and fuzzy set properties
• An analysis of fuzzy inference and approximate reasoning, along with the concepts of containment and entailment and relations between fuzzy subsets

Perfect for mid-level and upper-level undergraduate and graduate students in electrical, mechanical, and computer engineering courses, Introduction to Fuzzy Logic covers topics included in many artificial intelligence, computational intelligence, and soft computing courses. Math students and professionals in a wide variety of fields will also significantly benefit from the material covered in this book.

Preface (1-11)

Acknowledgements ( 1 )

About the Author ( 1 )

Introduction

Chapter 1 A Brief Introduction and History 1

Introduction 1

Models of Human Reasoning 1

The Early Foundation 2

Building On The Past - From Those Who Laid The Foundation 3

A Learning and Reasoning Taxonomy 4

Rote Learning 4

Learning With a Teacher 5

Learning by Example 5

Analogical or Metaphorical Learning 6

Learning by Problem Solving 6

Learning By Discovery 7

Crisp and Fuzzy Logic 7

Starting To Think Fuzzy 7

History Revisited - Early Mathematics 9

Foundations of Fuzzy Logic 9

Fuzzy Logic And Approximate Reasoning 9

Non-Monotonic Reasoning 11

Sets and Logic 12

Classical Sets 12

Fuzzy Subsets 13

Fuzzy Membership Functions 14

Expert Systems 16

Summary 17

Review questions 17

Chapter 2 A Review of Boolean Algebra 19

Introduction to crisp logic and Boolean Algebra 19

Introduction to algebra 20

Postulates 20

Theorems 23

Getting some practice 24

Getting to work 24

Boolean Algebra 24

Implementation 28

Logic minimization 29

Algebraic Means 29

Karnaugh Maps 30

Applying the K-map 30

2 Variable K-Maps 31

3 Variable K-Maps 32

4 Variable K-Maps 33

Going Backwards 33

Don’t Care Variables 35

Summary 37

Review questions 37

Chapter 3 Crisp Sets and Sets and More Sets 38

Introducing the Basics 38

Introduction to Classic Sets and Set Membership 41

Classic Sets 41

Set Membership 41

Basic Classic Crisp Set Properties 45

Exploring Sets and Set Membership 46

Fundamental Terminology 47

Elementary Vocabulary 47

Classical Set Theory and Operations 49

Classic Set Logic 49

Basic Classical Crisp Set Properties 50

Basic Crisp Applications – A First Step 57

Summary 59

Review questions 60

Chapter 4 Fuzzy Sets and Sets and More Sets 61

Introducing Fuzzy 61

Early Mathematics 62

Foundations of Fuzzy Sets Logic 62

Introducing the Basics 64

Introduction to Fuzzy Sets and Set Membership 66

Fuzzy Subsets and Fuzzy Logic 66

Fuzzy Membership Functions 68

Fuzzy Set Theory and Operations 71

Fundamental Terminology 71

Basic Fuzzy Set Properties and Operations 72

Basic Fuzzy Applications – A First Step 83

A Crisp Activity revisited 83

Fuzzy Imprecision and Membership Functions 86

Linear Membership Functions 87

Curved Membership Functions 90

Summary 95

Review questions 96

Chapter 5 What do You Mean by That? 97

Language, Linguistic Variables, Sets And Hedges 97

Symbols And Sounds To Real World Objects 99

Crisp Sets a Second Look 99

Fuzzy Sets a Second Look 103

Linguistic Variables 103

Membership Functions 105

Hedges 106

Summary 110

Review questions 111

Chapter 6 If There Were Four Philosophers 112

Fuzzy Inference And Approximate Reasoning 112

Equality 113

Containment And Entailment 116

Relations Between Fuzzy Subsets 119

Union and Intersection 119

Conjunction and Disjunction 121

Conditional Relations 125

Composition Revisited 127

Max-Min Composition 128

Max-Product Composition 130

Inference In Fuzzy Logic 137

Summary 140

Review questions 141

Chapter 7 So How Do I Use This Stuff? 142

Introduction 142

Fuzzification and Defuzzification 143

Fuzzification 143

Defuzzification 146

Fuzzy Inference Revisited 147

Fuzzy Implication 148

Fuzzy Inference - Single Premise 149

Max Criterion 150

Mean of Maximum 151

Center of Gravity 152

Fuzzy Inference - Multiple Premises 153

Getting to work - Fuzzy Control and Fuzzy Expert Systems 154

Membership Functions 158

System Behavior 159

Defuzzification Strategy 160

Membership Functions 162

System Behavior 163

Defuzzification Strategy 164

Summary 165

Review questions 166

Chapter 8 I Can Do This Stuff !!! 167

Introduction 167

Applications 167

Design Methodology 168

Executing a Design Methodology 169

Summary 172

Review questions 172

Chapter 9 Moving to Threshold Logic !!! 173

Introduction 173

Threshold Logic 173

Executing a Threshold Logic Design 174

Designing an AND Gate 175

Designing an OR Gate 175

Designing a Fundamental Boolean Function 176

The Downfall of Threshold Logic Design 179

Summary 180

Review Questions 181

Chapter 10 Moving to Perceptron Logic !!! 182

Introduction 182

The Biological Neuron 183

Dissecting the Biological Neuron 184

The Artificial Neuron – A First Step 185

The Perceptron – The Second Step 189

The Basic Perceptron 190

Single and Multilayer Perceptron 192

Bias and Activation Function 193

Learning with Perceptrons – First Step 196

Learning with Perceptrons – The Learning Rule 197

Learning with Perceptrons –Second Step 200

Path of the Perceptron Inputs 201

Testing of the Perceptron 203

Summary 204

Review Questions 205

Appendix A Requirements and Design Specifications 207

Introduction 207

Identifying the requirements 209

Formulating the requirements specification 211

The Environment 212

Characterizing External Entities 212

The System 213

Characterizing the System 214

System Inputs And Outputs 214

Functional View 215

Operational View 215

Technological View 215

Safety, Security, And Reliability 216

The System Design Specification 223

The System 225

Quantifying the System 225

System Requirements Versus System Design Specifications 335

Appendix B Introduction to UML 237

Introduction 237

Use Cases 238

Writing a Use Case 240

Class Diagrams 241

Class Relationships 242

Inheritance or Generalization 242

Interface 243

Containment 243

Aggregation 243

Composition 244

Dynamic Modeling with UML 245

Interaction Diagrams 245

Call and Return 246

Create and Destroy 246

Send 247

Sequence diagrams 247

Fork and join 248

Branch and merge 249

Activity diagram 250

State chart diagrams 251

Events 251

State Machines and State Chart Diagrams 252

UML State Chart Diagrams 252

Transitions 253

Guard Conditions 253

Composite States 254

Sequential States 254

History States 255

Concurrent Substates 255

Data Source / Sink 256

Data Store 256

Preparing for Test 258

Thinking Test 258

Examining the Environment 259

Test Equipment 259

The Eye Diagram 260

Generating the Eye Diagram 260

Interpreting the Eye Diagram 261

Back of the Envelope Examination 262

A First Step Check List 262

Routing and Topology 263

Summary 263

Bibliography

Index

James K. Peckol, PhD, is Principal Lecturer Emeritus in the Department of Electrical and Computer Engineering at the University of Washington in Seattle. He has over 50 years of experience in engineering and education in the fields of software, digital, medical, and embedded systems design and development.