You are previewing Novel Developments in Granular Computing: Applications for Advanced Human Reasoning and Soft Computation.
O'Reilly logo
Novel Developments in Granular Computing: Applications for Advanced Human Reasoning and Soft Computation

Book Description

Novel Developments in Granular Computing: Applications for Advanced Human Reasoning and Soft Computation analyzes developments and current trends of granular computing, reviewing the most influential research and predicting future trends. This book not only presents a comprehensive summary of existing practices, but enhances understanding on human reasoning.

Table of Contents

  1. Copyright
  2. Editorial Advisory Board
  3. Preface
    1. 1. GRANULAR COMPUTING
    2. 2. RECENT DEVELOPMENTS IN GRANULAR COMPUTING
      1. 2.1 Philosophic and Fundamental Views of Granular Computing
      2. 2.2 Human-Centered and Fuzzy Information Processing
      3. 2.3 Rough-Granular Computing
      4. 2.4 Dominance-Based Rough Set Approach
      5. 2.5 Other Important Research Directions
    3. 3. THE MOST CITED GRANULAR COMPUTING PAPERS AND CALL FOR CHAPTERS
    4. 4. CHAPTER SUMMARY
    5. REFERENCES
    6. ADDITIONAL READING
  4. Acknowledgment
  5. 1. Human-Inspired Granular Computing
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. GRANULAR COMPUTING AS HUMAN-INSPIRED PROBLEM SOLVING
    4. 3. HUMAN-ORIENTED AND MACHINE-ORIENTED STUDIES
    5. 4. GRANULAR COMPUTING FOR HUMANS
    6. 5. GRANULAR COMPUTING FOR MACHINES
    7. 6. THE TRIARCHIC THEORY OF GRANULAR COMPUTING
      1. 6.1. Granular Structures as Multiple Hierarchies
      2. 6.2. The Granular Computing Triangle
        1. Philosophy
        2. Methodology
        3. Computation
    8. 7. CONCLUDING REMARKS
    9. REFERENCES
  6. 2. Discovery of Process Models from Data and Domain Knowledge: A Rough-Granular Approach
    1. ABSTRACT
    2. INTRODUCTION: WISDOM TECHNOLOGY (WISTECH)
    3. OPTIMIZATION IN DISCOVERY OF COMPOUND GRANULES
        1. Definition 1
      1. Definition 2
        1. Example 1
        2. Example 2
    4. GRANULATION BY DECOMPOSITION AND CONTEXTUAL GRANULATION
      1. Example 3
    5. RGC IN PROCESS MINING
      1. Introduction
      2. Function Approximation and Rough Integral
        1. Example 4
      3. Interactions of Granules and Reasoning about Changes
      4. Example 5
      5. Example 6: Rules about Changes
    6. CONCLUSION
    7. ACKNOWLEDGMENT
    8. REFERENCES
    9. ENDNOTES
  7. 3. Supervised and Unsupervised Information Granulation: A Study in Hyperbox Design
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. FUZZY MIN-MAX CLASSIFICATION
        1. Hyperbox Expansion
      1. Overlap Test
      2. Hyperbox Contraction
    4. 3. INHERENT LIMITATIONS OF THE FUZZY MIN-MAX CLASSIFICATION
    5. 4. EXCLUSION/INCLUSION FUZZY CLASSIFICATION NETWORK (EFC)
    6. 5. NUMERICAL EXAMPLE
    7. 6. FROM FUZZY CLUSTERING TO HYPERBOX INFORMATION GRANULES
    8. 7. THE CLUSTERING ALGORITHM: DETAILED CONSIDERATIONS
    9. 8. CONCLUSION
    10. ACKNOWLEDGMENT
    11. REFERENCES
  8. 4. On Characterization of Relation Based Rough Set Algebras
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. GENERALIZED CRISP ROUGH SET MODELS
      1. 2.1. Construction of Crisp Approximation Operators
      2. 2.2. Axiomatic Characterization of Generalized Approximation operators
        1. Definition 1
        2. Definition 2
        3. Theorem 1
        4. Definition 3
        5. Theorem 2
        6. Theorem 3. (Yao, 1998a)
        7. Theorem 4. (Yao, 1998a)
        8. Theorem 5. (Yao, 1998a)
    4. 3. ROUGH FUZZY SET MODELS
      1. 3.1. Construction of Rough Fuzzy Approximation operators
      2. 3.2. Axiomatic Characterization of Rough Fuzzy approximation operators
        1. Definition 4
        2. Theorem 6. (Wu and Zhang, 2004)
        3. Definition 5
        4. Theorem 7. (Wu and Zhang, 2004)
        5. Theorem 8. (Wu and Zhang, 2004)
        6. Theorem 9. (Wu and Zhang, 2004)
    5. 4. FUZZY ROUGH SET MODELS DETERMINED BY A TRIANGULAR NORM
      1. 4.1. Construction of Fuzzy Rough Approximation operators
      2. 4.2. Axiomatic Characterization of Fuzzy Rough Approximation operators
        1. Theorem 11. (Wu et al., 2005)
        2. Definition 6
        3. Theorem 12. (Wu et al., 2005)
        4. Theorem 13. (Wu et al., 2005)
        5. Theorem 14. (Wu et al., 2005)
        6. Theorem 15. (Wu et al., 2005)
      3. 4.3. Relationship Between Fuzzy Rough Set Algebras and Fuzzy Topological Spaces
        1. Theorem 16
        2. Theorem 17
    6. 5. CONCLUSION
    7. ACKNOWLEDGMENT
    8. REFERENCES
  9. 5. A Top-Level Categorization of Types of Granularity
    1. ABSTRACT
    2. INTRODUCTION
    3. BACKGROUND
      1. Related Works
        1. Modelling: Subject Domain Semantics and Ontology
        2. Granular Computing
    4. ANALYSIS OF DIFFERENT EMPHASES REGARDING GRANULARITY
      1. Emphasis on Entity Types and Their Instances
        1. Example 1
      2. Emphasis on Relation between Entities and Levels
      3. Emphasis on the perspective and Criteria for Granulation
      4. Emphasis on Formal representation
    5. MAIN DIFFERENCES CONCERNING APPROACHES TOWARD GRANULARITY
    6. TAXONOMY OF TYPES OF GRANULARITY
      1. Overview of the Top-Level Taxonomy
      2. Characteristics in Detail
      3. Characteristics of the Eight leaf types
        1. saoG
        2. samG
        3. sgpG
        4. sgrG
        5. nrG
        6. nfG
        7. nacG
        8. nasG
    7. SAMPLE USAGE OF THE TYPES OF GRANULARITY IN MODELING AND IMPLEMENTATION
      1. Granular Perspectives and Their Type of Granularity
        1. Content of a Level with Arbitrary Scales
      2. Non-Scale-Dependent Content of a Level
        1. Example 4
      3. Simplifying implementation by Using the Types of Granularity
    8. FUTURE TRENDS
    9. CONCLUSION
    10. REFERENCES
    11. ENDNOTES
  10. 6. From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval, p-Box, and Fuzzy Uncertainty
    1. ABSTRACT
    2. 1. FORMULATION OF THE PROBLEM
      1. 1.1. Need for Data Processing
      2. 1.2. Measurement Uncertainty: From Probabilities to Intervals
      3. 1.3. Case of Fuzzy uncertainty and its Reduction to interval uncertainty
      4. 1.4. outline
    3. 2. INTERVAL COMPUTATIONS: BRIEF REMINDER
      1. 2.1. Interval Computations: Main Idea
      2. 2.2. From Main Idea to Actual Computer Implementation
      3. 2.3. Sometimes, We Get Excess Width
      4. 2.4. Reason for Excess Width
    4. 3. CONSTRAINT-BASED SET COMPUTATIONS
      1. 3.1. Main Idea
      2. 3.2. From Main Idea to Actual Computer Implementation
      3. 3.3. First Example: Computing the Range of x — x
      4. 3.4 Second Example: Computing the Range of x - x2
      5. 3.5. How to Compute rik,
      6. 3.6. Limitations of This Approach
      7. 3.7. Cases When This Approach is Applicable
      8. 3.8. Example: Estimating Variance Under Interval Uncertainty
        1. Comment
      9. 3.9. Other Statistical Characteristics
      10. 3.10. Systems of Ordinary Differential Equations (ODEs) under Interval Uncertainty
      11. 3.11 example
      12. 3.12. Solving Systems of Ordinary Differential Equations (ODEs) under Interval Uncertainty
      13. 3.13. Other Possible Cases When Our Approach Is Efficient
        1. Comment
      14. 3.14. Additional Advantage of Our Technique: Possibility to Take Constraints into Account
      15. 3.15. Toy Example with a Constraint
    5. 4. POSSIBLE EXTENSION TO P-BOXES AND CLASSES OF PROBABILITY DISTRIBUTIONS
      1. 4.1. Classes of Probability Distributions and p-Boxes: A Reminder
      2. 4.2. Propagating p-Box Uncertainty via Computations: A problem
      3. 4.3. Idea
      4. 4.4. From Idea to Computer Implementation
      5. 4.5. For p-Boxes, We Need Further Improvements to Make This Method practical
    6. ACKNOWLEDGMENT
    7. REFERENCES
  11. 7. Granular Computing Based Data Mining in the Views of Rough Set and Fuzzy Set
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. BASIC CONCEPTS OF RELATED THEORIES
      1. Domain-Oriented Data-Driven Data Mining (3DM)
      2. Granular Computing
      3. Rough Set Theory and Fuzzy Set Theory
        1. Rough Set Theory
        2. Fuzzy Set Theory
    4. 3. GRANULAR COMPUTING BASED DATA MINING
      1. Two Granular Models for Processing Incomplete Information Systems
        1. Some Extended Models of Classical Rough Set Theory
        2. Granular Computing Based on Tolerance Relation
      2. Algorithm 1: Granular Space Generation for Incomplete Information Systems (Wang et al., 2005)
        1. Granular Computing Based on Fuzzy-Clustering
      3. Uncertainty in Covering Based Granular Computing
        1. Uncertainty Measure of Pawlak's Rough Set
        2. Uncertainty Measure of Covering Based Rough Set
        3. Relationship of Covering Approximation Space and Its Transformed Pawlak's Approximation Space
      4. Attribute Reduction Based on Granular Computing
        1. Encoding Granules with Bitmap Technique
        2. Attribute Reduction Based on Granular Computing
      5. Rule Generation Based on Granular Computing
        1. A Rule Generation Algorithm Based on Granular Computing
        2. A Self-Learning Algorithm Based on Granular Computing
    5. 4. CONCLUSION
    6. 5. ACKNOWLEDGMENT
    7. REFERENCES
  12. 8. Near Sets in Assessing Conflict Dynamics within a Perceptual System Framework
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. APPROACHES TO ASSESSING CONFLICT DYNAMICS: AN OVERVIEW
      1. 2.1. Conflict Model
      2. 2.2. Conflict Graphs
      3. 2.3. Approximation spaces
      4. 2.4. Risk patterns
    4. 3. MODELLING CONFLICT SITUATIONS AS PERCEPTUAL INFORMATION SYSTEMS
      1. 3.1. Perceptual Information systems: Formal Definition
      2. 3.2. Relations, Partitions, and Classes
        1. Definition 1. Indiscernibility Relation (Pawlak, 1981a)
        2. Definition 2. Weak Indiscernibility Relation (Ewa Orlowska, 1998)
        3. Definition 3. Weak Tolerance Relation
        4. Definition 4
      3. 3.3. Nearness Relations and Tolerance perceptual Near sets
        1. Definition 5. Nearness Relation (Peters and Wasilewski, 2009)
        2. Definition 6. Weak Nearness Relation (Peters and Wasilewski, 2009)
        3. Definition 7. Weak Tolerance Nearness Relation (Peters, 2009)
        4. Definition 8. Tolerance Perceptual Near Sets (Peters, 2009)
        5. Definition 9. Tolerance Nearness Measure
    5. 4. ASSESSING CONFLICT DYNAMICS WITH NEAR SETS: ILLUSTRATION
        1. Example 1. Comparing Perceptual ISs Representing Conflict
          1. Tolerance Perceptual Near Sets
          2. Tolerance Nearness Measure
        2. Example 2. Comparing Perceptual ISs Representing Conflict
          1. Tolerance Perceptual Near Sets
          2. Tolerance Nearness Measure
    6. 5. CONCLUSION
    7. ACKNOWLEDGMENT
    8. REFERENCES
    9. ENDNOTE
  13. 9. Rule Extraction and Rule Evaluation Based on Granular Computing
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. PRELIMINARIES
    4. 3. RULE-EXTRACTING APPROACH UNDER DYNAMIC GRANULATION
      1. 3.1. Positive Approximation and Rule Extraction
        1. Definition 3.1
        2. Theorem 3.1
        3. Theorem 3.2
      2. Algorithm 3.1
      3. 3.2. Converse Approximation and Rule Extraction
        1. Definition 3.2
        2. Theorem 3.3
        3. Example 3.2
        4. Theorem 3.4
        5. Definition 3.3
        6. Theorem 3.5
        7. Theorem 3.6
      4. Algorithm 3.2
        1. Example 3.3
        2. Example 3.4
    5. 4. EVALUATING THE DECISION PERFORMANCE OF A DECISION TABLE
      1. 4.1. Decision Rule and Knowledge Granulation in Decision Tables
        1. Definition 4.1
        2. Definition 4.2
        3. Definition 4.3
        4. Definition 4.4
        5. Lemma 4.1
        6. Lemma 4.2
        7. Theorem 4.1
      2. 4.2. Limitations of Classical Measures for Decision Tables
        1. Example 4.1.
        2. Property 4.1
      3. 4.3. Evaluation of the Decision performance of a Rule set
        1. Definition 4.5
        2. Theorem 4.2. (Extremum)
        3. Example 4.2
        4. Theorem 4.3.
        5. Theorem 4.4
        6. Definition 4.6
        7. Theorem 4.5 (Extremum)
        8. Theorem 4.6
        9. Theorem 4.7
        10. Example 4.3
        11. Theorem 4.8. (Extremum)
        12. Example 4.4 (Continued from Example 4.3)
        13. Theorem 4.9
        14. Corollary 4.1
        15. Theorem 4.10
        16. Theorem 4.11
        17. Corollary 4.2
        18. Corollary 4.3
    6. 5. CONCLUSION
    7. ACKNOWLEDGMENT
    8. REFERENCES
  14. 10. Granular Models: Design Insights and Development Practices
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. THE CLUSTER-BASED REPRESENTATION OF THE INPUT-OUTPUT MAPPINGS
    4. 3. CONTEXT-BASED CLUSTERING IN THE DEVELOPMENT OF GRANULAR MODELS
    5. 4. GRANULAR NEURON AS A GENERIC PROCESSING ELEMENT IN GRANULAR NETWORKS
    6. 5. ARCHITECTURE OF GRANULAR MODELS BASED ON CONDITIONAL FUZZY CLUSTERING
    7. 6. REFINEMENTS OF GRANULAR MODELS
      1. 6.1. Bias term of Granular neurons
      2. 6.2. Refinement of the Contexts
    8. 7. INCREMENTAL GRANULAR MODELS
      1. 7.1. The principle of Incremental fuzzy Model and Its Design and Architecture
    9. 8. CONCLUSION
    10. REFERENCES
  15. 11. Semantic Analysis of Rough Logic
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. BASIC CONCEPT OF ROUGH LOGIC
      1. Definition 1
    4. 3. TRUTH VALUES OF ROUGH LOGICAL FORMULAE
      1. Definition 2
    5. 4. SEMANTIC MODEL OF ROUGH LOGIC
    6. 5. Λ-SATISFIABILITY OF ROUGH LOGICAL FORMULAE
      1. Definition 3
    7. 6. RELATED OPERATIONS OF SEMANTICS BASED ON THE ROUGH LOGIC
      1. Definition 4
        1. Example 1
      2. 6.1. λ -inclusion Operation
      3. 6.2. λ -Closeness Operation
    8. 7. RELATED PROPERTIES OF SEMANTICS OF ROUGH LOGIC
      1. Property 1
      2. Property 2
      3. Property 3
      4. Property 4
      5. Property 5
      6. Property 6
      7. Property 7
      8. Property 8
      9. Property 9
      10. Property 10
      11. Property 11
      12. Property 12
        1. Proof
    9. 8. NORMAL FORMS BASED ON MEANING OF ROUGH LOGICAL FORMULAE
      1. 8.1. Disjunction Normal Form
      2. 8.2. Conjunction Normal Form
      3. 8.3. Skolem Clause Form
    10. 9. SEMANTIC REASONING OF ROUGH LOGIC
      1. Proof
      2. 9.1. Deductive Reasoning of semantics
        1. Example 2
      3. Proof
      4. 9.2. λ -Precision Reasoning of Semantics
        1. Theorem
        2. Definition 13
        3. Definition 14
        4. Example 3
      5. 9.3. Lock Reasoning of Semantics
        1. Definition 15
        2. Example 4
    11. 10. APPLICATIONS OF SEMANTICS OF ROUGH LOGIC
      1. Example 5
    12. 11. PERSPECTIVE OF STUDYING SEMANTICS OF ROUGH LOGIC
    13. REFERENCES
  16. 12. Rough Entropy Clustering Algorithm in Image Segmentation
    1. ABSTRACT
    2. 1. INTRODUCTION
      1. 1.1. Motivation
      2. 1.2. Nomenclature
      3. 1.3. Paper Organization
    3. 2. IMAGE SEGMENTATION AND CLUSTERING
      1. 2.1. Overview of Segmentation Methods
      2. 2.2. Image Types and Attributes
      3. 2.3. Clustering Techniques
    4. 3. ROUGH SETS AND ROUGH ENTROPY
      1. 3.1. Rough Set Theory Essentials
      2. 3.2. Entropy Measure
      3. 3.3. Rough Entropy Measure
    5. 4. EVOLUTIONARY ALGORITHMS
      1. 4.1. General Remarks
      2. 4.2. Evolutionary Algorithm Performance
        1. Quantitative Measure β-Index
        2. Quantitative Measure KM: k-Means Partition Measure
        3. Quantitative Measure wV: Within Class Variance Measure
    6. 5. ROUGH ENTROPY CLUSTERING ALGORITHM
      1. 5.1. Granular Rough Entropy Thresholding
      2. 5.2. 1D GMRET Algorithm
      3. 5.3. 2D GMRET Algorithm
      4. 5.4. RECA Algorithm
    7. 6. EXPERIMENTS
      1. 6.1. Experimental Setup
      2. 6.2. Experimental Results
        1. Lenna Images
        2. IKONOS Images
        3. Nature Image
        4. Building Image
    8. DISCUSSION AND CONCLUSION
    9. ACKNOWLEDGMENT
    10. REFERENCES
  17. 13. Modelling Classification by Granular Computing
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. GRANULAR STRUCTURES
      1. 2.1 Information Tables and a Logic Language
        1. Definition 1
        2. Definition 2
      2. 2.2 Granules
        1. Definition 3
      3. 2.3 Levels and Granulations
        1. Definition 4
      4. 2.4 Hierarchies
      5. 2.5 Granule Networks
    4. 3. MODELLING CLASSIFICATION
      1. 3.1 Classifying a Granule
        1. Definition 5
      2. 3.2 Classifying a Granulation
        1. Definition 6
      3. 3.3 Modelling Classification as a Search in a Granule Network
        1. 3.3.1 Top-Down Algorithms
        2. 3.3.2 Bottom-Up Algorithms
    5. 4. INTERACTIVE TOP-DOWN CLASSIFICATION AND EXHAUSTIVE BOTTOM-UP CLASSIFICATION
    6. 5. CONCLUSION
    7. REFERENCES
  18. 14. Discovering Perceptually Near Granules
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. PERCEPTUAL OBJECTS, PERCEPTUAL GRANULES, AND PERCEPTUAL SYSTEMS
      1. Definition 2.1: Perceptual Granule
      2. 2.1 Perceptual Granules as Dual Chopped Lattices
        1. Observation 2.1
      3. 2.2 Perceptual Systems
      4. Definition 2.1: Perceptual System (Peters,(2007b), Peters and Wasilewski,(2009), Peters,(2010))
    4. 3. NEARNESS RELATIONS
      1. Definition 3.1: Indiscernibility Relation (pawlak, 1981)
      2. Definition 3.2: Weak Indiscernibility Relation (Ortowska, 1998)
      3. Definition 3.2: Weak Tolerance Relation
      4. Definition 3.3: Nearness Relation (Peters and Wasilewski,(2009))
      5. Definition 3.4: Weak Nearness Relation (Peters and Wasilewski (2009))
      6. Definition 3.4: Weak Tolerance Nearness Relation
    5. 4. NEAR PERCEPTUAL GRANULES
        1. Example 4.1 Near Images
        2. Example 4.2 Near Satellite Images
    6. 5. PERCEPTUAL NEAR SETS
      1. Definition 5.1: Perceptual Near Sets (Peters and Wasilewski (2009))
        1. Theorem 5.1 (Peters and Wasilewski (2009))
        2. Remark 5.1
        3. Example 5.1
      2. Definition 5.2: Weak Perceptual Near Sets (Peters and Wasilewski,(2009))
        1. Example 5.2
      3. Definition 5.3: Tolerance Near Sets (Peters,(2009a), Peters,(2010)).
        1. Example 5.2
    7. 6. PROBABILISTICALLY NEAR COMMUNICATING SYSTEMS
      1. 6.1 Basic Approach: Probabilistic Method
      2. Definition 6.1: Probability Space (Mitzenmacher, Upfal (2005))
      3. Definition 6.2: Probabilistic Method
      4. Definition 6.3: Conditional probability
      5. 6.2 Probability Space for a Communicating System
        1. Proposition 6.1
      6. 6.3 Affinities Between Communicating Systems
      7. Definition 6.4: Probabilistic Nearness Relation
        1. Proposition 6.2
        2. Example 6.1: Near Communicating Systems
    8. 7. ROUGH SETS AS NEAR PERCEPTUAL GRANULES
      1. Proposition 7.1 (Peters,(2007d))
      2. Proposition 7.2 (Peters,(2007d))
    9. 8. CONCLUSION
    10. ACKNOWLEDGMENT
    11. REFERENCES
    12. ENDNOTE
  19. 15. Granular Computing in Object-Oriented Software Development Process
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. PRINCIPLES OF GRANULAR COMPUTING
    4. 3. OBJECT-ORIENTED SOFTWARE ENGINEERING PROCESS
    5. 4. GRANULAR COMPUTING IN OBJECT-ORIENTED SOFTWARE ENGINEERING PROCESS
      1. 4.1. Granular Computing in Object-Oriented Requirement Analysis
      2. 4.2. Granular Computing in object-oriented system Analysis
      3. 4.3. Granular Computing in Object-Oriented system Design
      4. 4.4. Granular Computing in Object-Oriented system implementation
      5. 4.5. Granular Computing in Object-Oriented software Testing
      6. 4.6. Granular Computing in Object-Oriented System Deployment Design
    6. 5. CONCLUDING REMARKS
    7. REFERENCES
  20. 16. Granular Computing in Formal Concept
    1. ABSTRACT
    2. BACKGROUND
    3. BASIC DEFINITION
      1. Definition 1.1
      2. Definition 1.2
      3. Definition 1.3
      4. Example 1.1
      5. Definition 1.4
    4. 2. IDEAL-FILTER GRANULE
      1. Definition 2.1
      2. Example 2.1
      3. Definition 2.2
      4. Theorem 2.1
        1. Proof
      5. Example 2.2
      6. Theorem 2.2
        1. Proof
      7. Definition 2.3
      8. Example 2.3
      9. Theorem 2.3
        1. Proof
      10. Example 2.4
    5. 3. CONGRUENCE GRANULE
      1. Definition 3.1
      2. Theorem 3.1
        1. Proof
      3. Example 3.1
      4. Theorem 3.2
      5. Example 3.2
      6. Theorem 3.3
        1. Proof
      7. Definition 3.2
      8. Example 3.3
      9. Definition 3.3
      10. Theorem 3.4
      11. Theorem 3.5
        1. Proof
      12. Example 3.4
    6. 4. TOLERANCE GRANULE
      1. Definition 4.1
      2. Example 4.1
      3. Definition 4.2
      4. Example 4.2
      5. Theorem 4.1
      6. Example 4.3
      7. Theorem 4.2
        1. Proof
      8. Example 4.5
      9. Definition 4.3
      10. Example 4.6
      11. Theorem 4.3
      12. Example 4.7
    7. 5. PROSPECT
    8. DEVELOPING TRENDS IN THE FUTURE
    9. CONCLUSION
    10. ACKNOWLEDGMENT
    11. REFERENCES
  21. 17. Granular Synthesis of Rule-Based Models and Function Approximation Using Rough Sets
    1. ABSTRACT
    2. INTRODUCTION
    3. BACKGROUND
    4. 3. RULE-BASED MODEL DEVELOPMENT AND FUNCTION APPROXIMATION
    5. COMPUTATIONAL RESULTS
      1. 4.1 Static System Modeling
      2. 4.2 Dynamic Systems Modeling
      3. 4.3 Non-Linear System Modeling
    6. 5. CONCLUSION
    7. ACKNOWLEDGMENT
    8. REFERENCES
  22. 18. A Genetic Fuzzy Semantic Web Search Agent Using Granular Semantic Trees for Ambiguous Queries
    1. ABSTRACT
    2. 1. INTRODUCTION
    3. 2. RELATED WORKS
    4. 3. GRANULAR SEMANTIC TREES FOR QUERIES DISAMBIGUATION
      1. Step 1: Fuzzification
      2. Step 2: Fuzzy Inference
      3. Step 3: Defuzzification
    5. 4. USING USERS' PREFERENCES TO PERFORM PERSONALIZED QUERIES DISAMBIGUATION
    6. 5. SEMANTIC SEARCH AGENT OPTIMIZED BY GENETIC ALGORITHMS
    7. 6. ANALYZING WEB PAGES USING GRANULAR SEMANTIC TREES
    8. 7. SIMULATIONS
    9. 8. CONCLUSION
    10. REFERENCES
  23. 19. Dominance-Based Rough Set Approach to Granular Computing
    1. ABSTRACT
    2. INTRODUCTION
    3. DOMINANCE-BASED ROUGH SET APPROACH
      1. Example Illustrating DRSA in the Context of Ordinal Classification
    4. PHILOSOPHICAL BASIS OF DRSA GRANULAR COMPUTING
    5. FUZZY SET EXTENSIONS OF THE DOMINANCE-BASED ROUGH SET APPROACH
    6. VARIABLE-CONSISTENCY DOMINANCE-BASED ROUGH SET APPROACH (VC-DRSA)
    7. ROUGH APPROXIMATIONS OF FUZZY SETS BASED ON THE PROPERTY OF MONOTONICITY
        1. Theorem 1
      1. Theorem 2
    8. CLASSICAL ROUGH SET AS A PARTICULAR CASE OF THE MONOTONIC ROUGH APPROXIMATION OF A FUZZY SET
        1. Example
        2. Theorem 3
    9. DOMINANCE-BASED ROUGH SET APPROACH TO CASE-BASED REASONING
        1. Theorem 4
    10. ALGEBRAIC STRUCTURES FOR DOMINANCE-BASED ROUGH SET APPROACH
      1. Bipolar de Morgan Brouwer-Zadeh Distributive Lattices
        1. Theorem 5
      2. Bipolar Nelson Algebra
        1. Theorem 6
      3. Bipolar Heyting Algebra
        1. Theorem 7
      4. Bipolar Double Stone Algebra
        1. Theorem 8
      5. Bipolar Three-Valued Łukasiewicz Algebra
        1. Theorem 9
      6. Bipolar Wajsberg Algebra
        1. Theorem 10
      7. An Algebra for Ordinal Classification
        1. Theorem 11
    11. BIPOLAR APPROXIMATION SPACE INDUCED FROM BIPOLAR QUASI BROUWER-ZADEH LATTICE
    12. CONCLUSION
    13. ACKNOWLEDGMENT
    14. REFERENCES
  24. Compilation of References
  25. About the Contributors