Quotient Space Based Problem Solving

Preface

Chapter 1. Problem Representations

1.1 Problem Solving

1.2 World Representations at Different Granularities

1.3 The Acquisition of Different Grain-Size Worlds

1.4 The Relation Among Different Grain Size Worlds

1.5 Property-Preserving Ability

1.6 Selection and Adjustment of Grain-Sizes

1.7 Conclusions

Chapter 2. Hierarchy and Multi-Granular Computing

2.1 The Hierarchical Model

2.2 The Estimation of Computational Complexity

2.3 The Extraction of Information on Coarsely Granular Levels

2.4 Fuzzy Equivalence Relation and Hierarchy

2.5 The Applications of Quotient Space Theory

2.6 Conclusions

Chapter 3. Information Synthesis in Multi-Granular Computing

3.1 Introduction

3.2 The Mathematical Model of Information Synthesis

3.3 The Synthesis of Domains

3.4 The Synthesis of Topologic Structures

3.5 The Synthesis of Semi-Order Structures

3.6 The Synthesis of Attribute Functions

Chapter 4. Reasoning in Multi-Granular Worlds

4.1 Reasoning Models

4.2 The Relation Between Uncertainty and Granularity

4.3 Reasoning (Inference) Networks (1)

4.4 Reasoning Networks (2)

4.5 Operations and Quotient Structures

4.6 Qualitative Reasoning

4.7 Fuzzy Reasoning Based on Quotient Space Structures

Chapter 5. Automatic Spatial Planning

5.1 Automatic Generation of Assembly Sequences

5.2 The Geometrical Methods of Motion Planning

5.3 The Topological Model of Motion Planning

5.4 Dimension Reduction Method

5.5 Applications

Chapter 6. Statistical Heuristic Search

6.1 Statistical Heuristic Search

6.2 The Computational Complexity

6.3 The Discussion of Statistical Heuristic Search

6.4 The Comparison between Statistical Heuristic Search and A∗ Algorithm

6.5 SA in Graph Search

6.6 Statistical Inference and Hierarchical Structure

Chapter 7. The Expansion of Quotient Space Theory

7.1 Quotient Space Theory in System Analysis

7.2 Quotient Space Approximation and Second-Generation Wavelets

7.3 Fractal Geometry and Quotient Space Analysis

7.4 The Expansion of Quotient Space Theory

7.5 Conclusions

Addenda A. Some Concepts and Properties of Point Set Topology

Addenda B. Some Concepts and Properties of Integral and Statistical Inference

References

Index