Chapter 1 Problem Representations 1
1.1. Problem Solving 1
1.1.1. Expert Consulting Systems 2
1.1.2. Theorem Proving 2
1.1.3. Automatic Programming 2
1.1.4. Graphical Representation 3
1.1.5. AND/OR Graphical Representation 3
1.2. World Representations at Different Granularities 5
1.2.1. The Model of Different Grain-Size Worlds 5
1.2.2. The Definition of Quotient Space 7
1.3. The Acquisition of Different Grain-Size Worlds 8
1.3.1. The Granulation of Domain 8
1.3.2. The Granulation by Attributes 9
1.3.3. Granulation by Structures 11
1.4. The Relation Among Different Grain Size Worlds 13
1.4.1. The Structure of Multi-Granular Worlds 13
1.4.2. The Structural Completeness of Multi-Granular Worlds 15
1.5. Property-Preserving Ability 21
1.5.1. Falsity-Preserving Principle 21
1.5.2. Quotient Structure 32
1.6. Selection and Adjustment of Grain-Sizes 32
1.6.1. Mergence Methods 33
Example 1.15 34
1.6.2. Decomposition Methods 34
1.6.3. The Existence and Uniqueness of Quotient Semi-Order 41
1.6.4. The Geometrical Interpretation of Mergence and Decomposition Methods 42
1.7. Conclusions 43
Chapter 2 Hierarchy and Multi-Granular Computing 45
2.1. The Hierarchical Model 45
2.2. The Estimation of Computational Complexity 48
2.2.1. The Assumptions 48
2.2.2. The Estimation of the Complexity Under Deterministic Models 49
2.2.3. The Estimation of the Complexity Under Probabilistic Models 54
2.3. The Extraction of Information on Coarsely Granular Levels 62
2.3.1. Examples 64
2.3.2. Constructing[f]Under Unstructured Domains 65
2.3.3. Constructing[f]Under Structured Domains 67
2.3.4. Conclusions 77
2.4. Fuzzy Equivalence Relation and Hierarchy 77
2.4.1. The Properties of Fuzzy Equivalence Relations 77
2.4.2. The Structure of Fuzzy Quotient Spaces 84
2.4.3. Cluster and Hierarchical Structure 86
2.4.4. Conclusions 88
2.5. The Applications of Quotient Space Theory 88
2.5.1. Introduction 88
2.5.2. The Structural Definition of Fuzzy Sets 90
2.5.3. The Robustness of the Structural Definition of Fuzzy Sets 95
2.5.4. Conclusions 102
2.6. Conclusions 102
Chapter 3 Information Synthesis in Multi-Granular Computing 105
3.1. Introduction 105
3.2. The Mathematical Model of Information Synthesis 106
3.3. The Synthesis of Domains 108
3.4. The Synthesis of Topologic Structures 109
3.5. The Synthesis of Semi-Order Structures 110
3.5.1. The Graphical Constructing Method of Quotient Semi-Order 110
3.5.2. The Synthesis of Semi-Order Structures 112
3.6. The Synthesis of Attribute Functions 117
3.6.1. The Synthetic Principle of Attribute Functions 117
3.6.2. Examples 120
3.6.3. Conclusions 126
Chapter 4 Reasoning in Multi-Granular Worlds 129
4.1. Reasoning Models 129
4.2. The Relation Between Uncertainty and Granularity 133
4.3. Reasoning(Inference)Networks(1) 135
4.3.1. Projection 138
4.3.2. Synthesis 140
4.3.3. Experimental Results 146
4.4. Reasoning Networks(2) 147
4.4.1. Modeling 151
4.4.2. The Projection of AND/OR Relations 152
4.4.3. The Synthesis of AND/OR Relations 155
4.4.4. Conclusion 160
4.5. Operations and Quotient Structures 160
4.5.1. The Existence of Quotient Operations 162
4.5.2. The Construction of Quotient Operations 164
4.5.3. The Approximation of Quotient Operations 172
4.5.4. Constraints and Quotient Constraints 176
4.6. Qualitative Reasoning 181
4.6.1. Qualitative Reasoning Models 182
4.6.2. Examples 182
4.6.3. The Procedure of Qualitative Reasoning 186
4.7. Fuzzy Reasoning Based on Quotient Space Structures 187
4.7.1. Fuzzy Set Based on Quotient Space Model 187
4.7.2. Fuzzified Quotient Space Theory 189
4.7.3. The Transformation of Three Different Granular Computing Methods 190
4.7.4. The Transformation of Probabilistic Reasoning Models 191
4.7.5. Conclusions 191
Chapter 5 Automatic Spatial Planning 193
5.1. Automatic Generation of Assembly Sequences 194
5.1.1. Introduction 194
5.1.2. Algorithms 195
5.1.3. Examples 199
5.1.4. Computational Complexity 202
5.1.5. Conclusions 204
5.2. The Geometrical Methods of Motion Planning 205
5.2.1. Configuration Space Representation 205
5.2.2. Finding Collision-Free Paths 206
5.3. The Topological Model of Motion Planning 207
5.3.1. The Mathematical Model of Topology-Based Problem Solving 208
5.3.2. The Topologic Model of Collision-Free Paths Planning 210
5.4. Dimension Reduction Method 216
5.4.1. Basic Principle 216
5.4.2. Characteristic Network 221
5.5. Applications 230
5.5.1. The Collision-Free Paths Planning for a Planar Rod 231
5.5.2. Motion Planning for a Multi-Joint Arm 237
5.5.3. The Applications of Multi-Granular Computing 242
5.5.4. The Estimation of the Computational Complexity 246
Chapter 6 Statistical Heuristic Search 249
6.1. Statistical Heuristic Search 251
6.1.1. Heuristic Search Methods 251
6.1.2. Statistical Inference 254
6.1.3. Statistical Heuristic Search 256
6.2. The Computational Complexity 259
6.2.1. SPA Algorithms 259
6.2.2. SAA Algorithms 262
6.2.3. Different Kinds of SA 264
6.2.4. The Successive Algorithms 266
6.3. The Discussion of Statistical Heuristic Search 267
6.3.1. Statistical Heuristic Search and Quotient Space Theory 267
6.3.2. Hypothesis I 268
6.3.3. The Extraction of Global Statistics 271
6.3.4. SA Algorithms 279
6.4. The Comparison between Statistical Heuristic Search and A* Algorithm 280
6.4.1. Comparison to A 280
6.4.2. Comparison to Other Weighted Techniques 283
6.4.3. Comparison to Other Methods 292
6.5. SA in Graph Search 294
6.5.1. Graph Search 294
6.5.2. AND/OR Graph Search 295
6.6. Statistical Inference and Hierarchical Structure 296
Chapter 7 The Expansion of Quotient Space Theory 299
7.1. Quotient Space Theory in System Analysis 299
7.1.1. Problems 300
7.1.2. Quotient Space Approximation Models 300
7.2. Quotient Space Approximation and Second-Generation Wavelets 303
7.2.1. Second-Generation Wavelets Analysis 303
7.2.2. Quotient Space Approximation 305
7.2.3. The Relation between Quotient Space Approximation and Wavelet Analysis 309
7.3. Fractal Geometry and Quotient Space Analysis 311
7.3.1. Introduction 311
7.3.2. Iterated Function Systems 311
7.3.3. Quotient Fractals 312
7.3.4. Conclusions 315
7.4. The Expansion of Quotient Space Theory 315
7.4.1. Introduction 315
7.4.2. Closure Operation-Based Quotient Space Theory 315
7.4.3. Non-Partition Model-Based Quotient Space Theory 320
7.4.4. Granular Computing and Quotient Space Theory 324
7.4.5. Protein Structure Prediction-An Application of Tolerance Relations 326
7.4.6. Conclusions 331
7.5. Conclusions 331
Addenda A:Some Concepts and Properties of Point Set Topology 333
Addenda B:Some Concepts and Properties of Integral and Statistical Inference 363
References 375
Index 381