Quotient Space Based Problem Solving:a Theoretical Foundation of Granular Computing

Preface
Chapter 1 Problem Representations
1.1.Problem Solving
1.1.1.Expert Consulting Systems
1.1.2.Theorem Proving
1.1.3.Automatic Programming
1.1.4.Graphical Representation
1.1.5.AND／OR Graphical Representation
1.2.World Representations at Diffrent Granularities
1.2.1.The Model of Different Grain—Size Worlds
1.2.2.The Definition of Quotient Space
1.3.The Acquisition of Difierent Grain—Size Worlds
1.3.1.The Granulation of Domain
1.3.2.The Granulation by Attributes
1.3.3.Granulation by Structures
1.4.The Relation Among Different Grain Size Worlds
1.4.1.The Structure of Multi—Granular Worlds
1.4.2.The Structural Completeness of Multi—Granular Worlds
1.5.Property—Preserving Ability
1.5.1.Falsity—Preserving Principle
1.5.2.Quotient Structure
1.6.Selection and Adjustment of Grain—Sizes
1.6.1.Mergence Methods
Example 1.15
1.6.2.Decomposition Methods
1.6.3.The Existence and Uniqueness of Quotient Semi—Order
1.6.4.The Geometrical Interpretation of Mergence and Decomposition Methods
1.7.Conclusions
Chapter 2 Hierarchy and Multi—Granular Computing
2.1.The Hierarchical Model
2.2.The Estimation of Computational Complexity
2.2.1.The Assumptions
2.2.2.The Estimation of the Complexity Under Deterministic Models
2.2.3.The Estimation of the Complexity Under Probabilistic Models
2.3.The Extraction of Information on Coarsely Granular Levels
2.3.1.Examples
2.3.2.Constructing （f） Under Unstructured Domains
2.3.3.Constructing （f） Under Structured Domains
2.3.4.Conclusions
2.4.Fuzzy Equivalence Relation and Hierarchy
2.4.1.The Properties of Fuzzy Equivalence Relations
2.4.2.The Structure of Fuzzy Quotient Spaces
2.4.3.Cluster and Hierarchical Structure
2.4.4.Conclusions
2.5.The Applications of Quotient Space Theory
2.5.1.Introduction
2.5.2.The Structural Definition of Fuzzy Sets
2.5.3.The Robustness of the Structural Definition of Fuzzy Sets
2.5.4.Conclusions
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 Domaias
3.4.The Synthesis of Topologic Structures
3.5.The Synthesis of Semi—Order Structures
3.5.1.The Graphical Constructing Method of Quotient Semi—Order
3.5.2.The Synthesis of Semi—Order Structures
3.6.The Synthesis of Attribute Functions
3.6.1.The Synthetic Principle of Attribute Functions
3.6.2.Examples
3.6.3.Conclusions
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）
4.3.1.Projection
4.3.2.Synthesis
4.3.3.Experimental Results
4.4.Reasoning Networks（2）
4.4.1.Modeling
4.4.2.The Projection of AND／OR Relations
4.4.3.The Synthesis of AND／OR Relations
4.4.4.Conclusion
4.5.Operations and Quotient Structures
4.5.1.The Existence of Quotient Operations
4.5.2.The Construction of Quotient Operations
4.5.3.The Approximation of Quotient Operations
4.5.4.Constraints and Quotient Constraints
4.6.Qualitative Reasoning
4.6.1.Qualitative Reasoning Models
4.6.2.Examples
4.6.3.The Procedure of Qualitative Reasoning
4.7.Fuzzy Reasoning Based on Quotient Space Structures
4.7.1.Fuzzy Set Based on Quotient Space Model
4.7.2.Fuzzified Quotient Space Theory
4.7.3.The Transformation of Three Different Granular Computing Methods
4.7.4.The Transformation of Probabilistic Reasoning Models
4.7.5.Conclusions
Chapter 5 Automatic Spatial Planning
5.1.Automatic Generation of Assembly Sequences
5.1.1.Introduction
5.1.2.Algorithms
5.1.3.Examples
5.1.4.Computational Complexity
5.1.5.Conclusions
5.2 The Geometrical Methods of Motion Planning
5.2.1.Configuration Space Representation
5.2.2.Finding Collision—Free Paths
5.3.The Topological Model of Motion Planning
5.3.1.The Mathematical Model of Topology—Based Problem Solving
5.3.2.The Topologic Model of Collision—Free Paths Planning
5.4.Dimension Reduction Method
5.4.1.Basic Principle
5.4.2.Characteristic Network
5.5.Applications
5.5.1.The Collision—Free Paths Planning for a Planar Rod
5.5.2.Motion Planning for a Multi—Joint Arm
5.5.3.The Applications of Multi—Granular Computing
5.5.4.The Estimation of the Computational Complexity
Chapter 6 Statistical Heuristic Search
6.1.Statistical Heuristic Search
6.1.1.Heuristic Search Methods
6.1.2.Statistical Inference
6.1.3.Statistical Heuristic Search
6.2.The Computational Complexity
6.2.1.SPA Algorithms
6.2.2.SAA Algorithms
6.2.3.Different Kinds of SA
6.2.4.The Successive Algorithms
6.3.The Discussion of Statistical Heuristic Search
6.3.1.Statistical Heuristic Search and Quotient Space Theory
6.3.2.Hypothesis I..
6.3.3.The Extraction of Global Statistics
6.3.4.SA Algorithms
6.4.The Comparison between Statistical Heuristic Search and A* Algorithm
6.4.1.Comparison to A*
6.4.2.Comparison to Other Weighted Techniques
6.4.3.Comparison to Other Methods
6.5.SA in Graph Search
6.5.1.Graph Search
6.5.2.AND／OR Graph Search
6.6.Statistical Inference and Hierarchical Structure
Chapter 7 The Expansion of Quotient Space therory
7.1.Quotient Space Theory in System Analysis
7.1.1.Problems
7.1.2.Quotient Space Approximation Models
7.2.Quotient Space Approximation and Second—Generation Wavelets
7.2.1.Second—Generation Wavelets Analysis
7.2.2.Quotient Space Approximation
7.2.3.The Relation between Quotient Space Approximation and Wavelet Analysis
7.3.Fractal Geometry and Quotient Space Analysis
7.3.1.Introduction
7.3.2.Iterated Function Systems
7.3.3.Quotient Fractals
7.3.4.Conclusions
7.4.The Expansion of Quotient Space Theory
7.4.1.Introduction
7.4.2.Closure Operation—Based Quotient Space Theory
7.4.3.Non—Partition Model—Based Quotient Space Theory
7.4.The Expansion of Quotient Space Theory
7.4.5.Protein Structure Prediction—An Application of Tolerance Relations
7.4.6.Conclusions
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