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Table of Contents Preface CHAPTER 1 Introduction 1. Digital communities and a fundamental quest for human-centric systems 2. A historical overview: towards a non-Aristotelian perspective of the world 3. Granular Computing 3.1. Sets and interval analysis 3.2. The role of fuzzy sets: a perspective of information granules 3.3. Rough sets 3.4. Shadowed sets 4. Quantifying information granularity: generality versus specificity 5. Computational Intelligence 6. Granular Computing and Computational Intelligence 7. Conclusions Exercises and problems Historical notes References CHAPTER 2 Notions and Concepts of Fuzzy Sets 1. Sets and fuzzy sets: a departure from the principle of dichotomy 2. Interpretation of fuzzy sets 3. Membership functions and their motivation 4. Fuzzy numbers and intervals 5. Linguistic variables 6. Conclusions Exercises and problems Historical notes References CHAPTER 3 Characterization of Fuzzy Sets 1. A generic characterization of fuzzy sets: some fundamental descriptors 2. Equality and inclusion relationships in fuzzy sets 3. Energy and entropy measures of fuzziness 4. Specificity of fuzzy sets 5. Geometric interpretation of sets and fuzzy sets 6. Granulation of information 7. Characterization of the families of fuzzy sets 8. Conclusions Exercises and problems Historical notes References CHAPTER 4 The Design of Fuzzy Sets 1. Semantics of fuzzy sets: some general observations 2. Fuzzy set as a descriptor of feasible solutions 3. Fuzzy set as a descriptor of the notion of typicality 4. Membership functions in the visualization of preferences of solutions 5. Nonlinear transformation of fuzzy sets 6. Vertical and horizontal schemes of membership estimation 7. Saaty?s priority method of pairwise membership function estimation 8. Fuzzy sets as granular representatives of numeric data 9. From numeric data to fuzzy sets 10. Fuzzy equalization 11. Linguistic approximation 12. Design guidelines for the construction of fuzzy sets 13. Conclusions Exercises and problems Historical notes References CHAPTER 5 Operations and Aggregations of Fuzzy Sets 1. Standard operations on sets and fuzzy sets 2. Generic requirements for operations on fuzzy sets 3. Triangular norms 3.1. Defining t-norms 3.2. Constructors of t-norms 4. Triangular conorms 4.1. Defining t- conorms 4.2. Constructors of t-conorms 5. Triangular norms as a general category of logical operators 6. Aggregation operations 6.1. Averaging operations 6.2. Ordered weighted averaging operations 6. 3. Uninorms and nullnorms 6.4. Symmetric sums 6.5. Compensatory operations 7. Fuzzy measure and integral 8. Negations 9. Conclusions Exercises and problems Historical notes References CHAPTER 6 Fuzzy Relations 1. The concept of relations 2. Fuzzy relations 3. Properties of the fuzzy relations 4. Operations on fuzzy relations 5. Cartesian product, projections and cylindrical extension of fuzzy sets 6. Reconstruction of fuzzy relations 7. Binary fuzzy relations 8. Conclusions Exercises and problems Historical notes References CHAPTER 7 Transformations of Fuzzy Sets 1. The extension principle 2. Compositions of fuzzy relations 2.1.sup-t composition 2.2. inf-s composition 2.3. inf-? composition 3. Fuzzy relational equations 3.1. Solutions to the estimation problem 3.2. Fuzzy relational system 3.3. Relation-relation fuzzy equations 3.4. Multi-input, single-output fuzzy relational equations 3.5. Solution of the estimation problem for equations with inf-s composition. 3.6. Solution of the inverse problem 3.7 Relation-relation fuzzy equations 3.8. Multi-input, single-output fuzzy relational equations 3.9. Solvability conditions for maximal solutions 4. Associative Memories 4.1. sup-t fuzzy associative memories 4.2. inf-s fuzzy associative memories 5. Fuzzy numbers and fuzzy arithmetic 5.1. Algebraic operations on fuzzy numbers 5.2. Computing with fuzzy numbers 5.3. Interval arithmetic and ?-cuts 5.4. Fuzzy arithmetic and the extension principle 5.5. Computing with triangular fuzzy numbers 6. Conclusions Exercises and problems Historical notes References CHAPTER 8 Generalizations and Extensions of Fuzzy Sets 1. Fuzzy sets of higher order 2. Rough fuzzy sets and fuzzy rough sets 3. Interval-valued fuzzy sets 4. Type-2 fuzzy sets 5. Shadowed sets as a three-valued logic characterization of fuzzy sets 5.1. Defining shadowed sets 5.2. The development of shadowed sets 6. Probability and fuzzy sets 7. Probability of fuzzy events 8. Conclusions Exercises and problems Historical notes References CHAPTER 9 Interoperability Aspects of Fuzzy Sets 1. Fuzzy set and its family of ?-cuts 2. Fuzzy sets and their interfacing with the external world 2.1. Encoding mechanisms 2.2. Decoding mechanisms 3. Encoding and decoding as an optimization problem of vector quantization 3.1. Fuzzy scalar quantization 3.2. Forming the mechanisms of the fuzzy quantization: beyond a winner-takes-all scheme 3.3. Coding and decoding with the use of fuzzy codebooks 4. Decoding of a fuzzy set through a family of fuzzy sets 4.1. Possibility and necessity measure in encoding of fuzzy data 4.2. The design of the decoder of fuzzy data 5. Taxonomy of data in structure description with shadowed sets 6. Conclusions Exercises and problems Historical notes References CHAPTER 10 Fuzzy Modeling: Principles and Methodology 1. The architectural blueprint of fuzzy models 2. Key phases of the development and use of fuzzy models 3. Main categories of fuzzy models: an overview 4. Verification and validation of fuzzy models 4.1. Verification of fuzzy models 4.2. Training, validation, and testing data in the development of fuzzy models 4.3. Validation of fuzzy models 5. Conclusions Exercises and problems Historical notes References CHAPTER 11 Rule-based Fuzzy Models 1. Fuzzy rules as a vehicle of knowledge representation 2. General categories of fuzzy rules and their semantic 3. Syntax of fuzzy rules 4. Basic Functional Modules: Rule base, Database, and Inference scheme 4.1. Input interface 4.2. Rule base 4.3. Main types of rule bases 4.4. Data base 4.5. Fuzzy inference 5. Types of Rule-Based Systems and Architectures 5.1. Linguistic fuzzy models 5.2. Functional (local) fuzzy models 5.3. Gradual fuzzy models 6. Approximation properties of fuzzy rule-based models 7. Development of Rule-Based Systems 7.1. Expert-driven development 7.2. Data-driven development 8. Parameter estimation procedure for functional rule-based systems 9. Design issues of rule-based systems ? consistency, completeness, and the curse of dimensionality 10. The curse of dimensionality in rule-based systems 11. Development scheme of fuzzy rule-based models 12. Conclusions Exercises and problems Historical notes References CHAPTER 12 From Logic Expressions to Fuzzy Logic Networks 1. Introduction 2. Main categories of fuzzy neurons 2. 1. Aggregative neurons 2.2. Referential (reference) neurons 3. Uninorm-based fuzzy neurons 3.1. Main classes of unineurons 3.2. Properties and characteristics of the unineurons 4. Architectures of logic networks 4.1. Logic processor in the processing of fuzzy logic functions: a canonical realization 4.2. Fuzzy neural networks with feedback loops 5. The development mechanisms of the fuzzy neural networks 5.1. The key design phases 5.2. Gradient-based learning schemes for the networks 6. Interpretation of the fuzzy neural networks 7. From fuzzy logic networks to Boolean functions and their minimization through learning 8. Interfacing the fuzzy neural network 9. Interpretation aspects ? a refinement of induced rule-based system 10. Reconciliation of perception of information granules and granular mappings 10.1. Reconciliation of perception of information granule 10.2. The optimization process 10.3. An application of the perception mechanism to fuzzy rule-based systems 10.4. Reconciliation of granular mappings 11. Conclusions Exercises and problems Historical notes References CHAPTER 13 Fuzzy Systems and Computational Intelligence 1. Computational Intelligence 2. Recurrent neurofuzzy systems 2.1. Recurrent neural fuzzy network model 2.2. Learning algorithm 3. Genetic fuzzy systems 4. Coevolutionary hierarchical genetic fuzzy system 5. Hierarchical collaborative relations 6. Evolving fuzzy systems 6.1.Functional fuzzy model 6.2. Evolving participatory learning algorithm 7. Conclusions Exercises and problems Historical notes References CHAPTER 14 Granular Models and Human Centric Computing 1. The cluster-based representation of the input ? output mappings 2. Context-based clustering in the development of granular models 3. Granular neuron as a generic processing element in granular networks 4. Architecture of granular models based on conditional fuzzy clustering 5. Refinements of granular models 5.1. Bias of granular neurons 5.2. Refinement of the contexts 6. Incremental granular models 6.1. The principle of incremental fuzzy model and its design and architecture 7. Human-centric fuzzy clustering 7.1. Fuzzy clustering with partial supervision 7.2. The development of the human-centric clusters 7.3. Proximity-based fuzzy clustering 8. Participatory learning in fuzzy clustering 9. Conclusions Exercises and problems Historical notes References CHAPTER 15 Emerging Trends in Fuzzy Systems 1. Relational ontology in information retrieval 1.1. Fuzzy relational ontological model 1.2. Information retrieval model and structure 1.3. Documents representation 1.4 Query representation 1.5. Information retrieval with relational ontological model 2. Multiagent fuzzy systems 2.1 Agents and multiagents 2.2 Electricity market 2.3. Genetic fuzzy system 3. Distributed fuzzy control 3.1 Resource allocation 3.2 Control systems and economy 3.3 Fuzzy market-based control 4. Conclusions Exercises and problems Historical notes References APPENDIX A Mathematical Prerequisites APPENDIX B Neurocomputing APPENDIX C Biologically Inspired Optimization
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