Granular, Fuzzy, and Soft Computing. A Volume in the Encyclopedia of Complexity and Systems Science, Second Edition

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Series Preface Volume Preface Soft and Nature-Inspired Computing Mathematical Fuzzy Sets and Rough Sets (Generalized) Topological Computing Granular Computing (GRC) Soft and Various Forms of Intelligent Computing Contents About the Editor-in-Chief About the Volume Editors Contributors Cooperative Multi-hierarchical Query Answering Systems Glossary Definition of the Subject Introduction Multi-hierarchical Decision System Cooperative Query Answering Future Directions Bibliography Dependency and Granularity in Data-Mining Definition of the Subject Introduction Contingency Table from Rough Sets Notations Contingency Table (m x n) Rank of Contingency Table (2 x 2) Preliminaries Independence When the Table Is 2 x 2 Rank of Contingency Table (m x n) Rank and Degree of Dependence Submatrix and Subdeterminant Rank and Subdeterminant Rank and Submatrix Determinantal Divisors Divisors and Degree of Dependence Subdeterminants and Degree of Dependence Elementary Divisors and Elementary Transformation Degree of Granularity and Dependence Conclusion Acknowledgments Bibliography Fuzzy Logic Glossary Definition of the Subject Introduction Conceptual Structure of Fuzzy Logic The Basics of Fuzzy Set Theory The Concept of Granulation The Concepts of Precisiation and Cointensive Precisiation The Concept of Cointension The Concept of a Generalized Constraint Constrained Variable Modalities of Generalized Constraints Possibilistic Constraint Probabilistic Constraint Veristic Constraint (Zadeh 1999, 2005a) Fuzzy Graph Constraint Usuality Constraint (Zadeh 1999, 2002) Primary Constraints Standard Constraints Generalized Constraint Language (GCL) (Zadeh 1999, 2005a) Deduction Principal Contributions of Fuzzy Logic Fuzzy Logic as the Basis for Generalization of Scientific Theories A Glimpse of What Lies Beyond Fuzzy Logic Bibliography Primary Literature Books and Reviews Fuzzy Probability Theory Glossary Fuzzy Set and Fuzzy Vector α-Level Set and Support σ-Algebra Definition Introduction Mathematical Environment Fuzzy Random Variables Fuzzy Probability Representation of Fuzzy Random Variables Application Fields Future Directions Bibliography Primary Literature Books and Reviews Fuzzy System Models Evolution from Fuzzy Rulebases to Fuzzy Functions Glossary Definition of the Subject Introduction Type 1 Fuzzy System Models of the Past Type 1 Fuzzy Rulebases Type 1 ``Special Fuzzy Functions´´ of the Recent Past Special Fuzzy Functions with LSE (SFF-LSE) Method Support Vector Machines for Regression Special Fuzzy Functions with SVM (FF-SVM) Method Future of Fuzzy System Models Basic Definitions General Structure of Type 2 Fuzzy System Models Discrete Interval Valued Type 2 Fuzzy Sets (DIVT2FS) Discrete Interval Valued Type 2 ``Special Fuzzy Functions´´, (SFF) Case Study Applications Daily Stock Price Dataset Income Prediction Dataset Desulphurization Dataset Experimental Design Sugeno-Yasukawa Models and Takagi-Sugeno Models Special Fuzzy Functions Models (SFFM) Experimental Model Results Conclusions and Future Directions Bibliography Primary Literature Books and Reviews On Genetic-Fuzzy Data-Mining Techniques Glossary Definition of the Subject and Its Importance Introduction Data Mining Fuzzy Sets Fuzzy Data Mining Genetic Algorithms Genetic-Fuzzy Data-Mining Techniques The Integrated Genetic-Fuzzy Problem for Items with a Single Minimum Support (IGFSMS) The Integrated Genetic-Fuzzy Problem for Items with Multiple Minimum Supports (IGFMMS) The Divide-and-Conquer Genetic-Fuzzy Problem for Items with a Single Minimum Support (DGFSMS) The Divide-and-Conquer Genetic-Fuzzy Problem for Items with Multiple Minimum Supports (DGFMMS) Multiobjective Genetic-Fuzzy Data Mining Future Directions Bibliography Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach Glossary Definition of the Subject and Its Importance Introduction: Granular Computing and Ordered Data Philosophical Basis of DRSA Granular Computing Dominance-Based Rough Set Approach Fuzzy Set Extensions of the Dominance-Based Rough Set Approach Variable-Consistency Dominance-Based Rough Set Approach (VC-DRSA) Dominance-Based Rough Approximation of a Fuzzy Set Monotonic Rough Approximation of a Fuzzy Set Versus Classical Rough Set Dominance-Based Rough Set Approach to Case-Based Reasoning An Algebraic Structure for Dominance-Based Rough Set Approach Conclusions Future Directions Bibliography Granular Computing, Information Models for Glossary Definition of the Subject Introduction Candidate Information Models The Relational Data Model (RDM) The Attribute-Based Data Model (ABDM) The Functional Data Model (FDM) The Extensible Markup Language (XML) The Resource Description Framework (RDF) The Web Ontology Language (OWL) Suitability of Information Models for Granular Computing Formalism and Theory Expressive Power Extensibility and Utility Future Directions Bibliography Introduction to Granular Computing Granular Computing from Rough Set and Generalized Theory Granular Computing in Database Theory Granular Computing in Social Networks Granular Computing and Fuzzy Set Theory Grid/Cloud Computing: A New Addition to GrC General Issues in Granular Computing Conclusion Bibliography Granular Computing and Modeling of the Uncertainty in Quantum Mechanics Glossary Definition of the Subject Introduction Quantum Postulates and Associated Propositions 0th Postulate of Quantum Mechanics 1st Postulate of Quantum Mechanics 2nd Postulate of Quantum Mechanics 3rd Postulate of Quantum Mechanics Dirac´s Bra and Ket Notations and the Realization of Wave Function Realization of q-Representation in Quantum Mechanics Fourier Transformation and p-Representation Quantum Uncertainty and Non-compatibility of Observables P and X Conclusion Future Directions Bibliography Philosophical Foundation for Granular Computing Glossary Definition of the Subject and Its Importance Introduction The Road to Granular Computing The Nature of Granule and Granule Computing The Nature of Granule The Nature of Granular Computing Granule Measurement Granular Structure Quotient View: Information Hiding Granular Computing and Ontologies Information Hiding and Emergence Granulation Provides a Unified View for Intelligent Problem Solving Re-examination of Existing Studies from a GrC Perspective Forming Intelligent Problem Solving Tasks in Terms of Concepts in Granular Computing Granular Computing and Data Mining/OLAP Relationship Between These Two Fields Granular Computing as a Basis for Data Mining and OLAP Relationship with Soft Computing and Natural Computing Relationship with Fundamental Issues of Computing and Complex Systems Problem Solving Granular Computing as Infrastructures for AI-Engineering Granular Computing and New Computational Models Temporal Aspects of Information Granules Today´s Big Challenges from Big Data Dynamic Granule: Capturing Dynamic Nature of Granules Temporal Aspects of Time-Varying Data Importance of Exploring Dynamic Granules at Big Data Era Other Related Developments Summary Future Directions Acknowledgments Bibliography Primary Literature Granular Computing: Practices, Theories, and Future Directions Glossary Definition of the Subject How Important Is Granular Computing? What Is Granular Computing? Illustration of Working Formal Models Introduction Classical Examples of Granulation Ancient Examples Modern Examples Further Examples Formal Models of Granulation First GrC Models and Ancient Examples Brief pre-GrC Historical Notes Second GrC Models and Modern Examples Third and Fourth GrC Models and Modern Examples Models for Further Examples Category Theory Based Models Overview of Early GrC Models Granular and Related Structures Integration - A Dual of Granulation Simple Examples of Integration Semantic Views Future Directions Bibliography Principles and Perspectives of Granular Computing Glossary Definition of the Subject of Its Importance Introduction Perspectives of Granular Computing Neighborhood System: A Mathematical Structure of Granular Computing Neighborhood System Approximation Rough Set System: An Equivalence Neighborhood System Partition Approximation Information Systems Quotient Space Fuzzy Set System: A Fuzzy Neighborhood System Information Granularity Approximation Granular Computing in Data Mining Granular Computing in Classification and Clustering Granular Computing in Association Mining A Framework for Granular Computing in Data Mining Granular Computing in Software Engineering User Requirements System Analysis System Design System Implementation Future Directions Bibliography Primary Literature Granular Computing System Vulnerabilities: Exploring the Dark Side of Social Networking Communities Glossary Definition of the Subject Introduction Background on Social Networking Communities Traditional Attacks Targeting Social Networking Communities Malware Propagation Worms Spyware Adware Spam Phishing New Attacks Against Social Networking Communities Social Spam Campaigns Misinformation Future Directions Bibliography Granular Model for Data Mining Glossary Definition of the Subject Introduction Granular Model: Syntax and Semantics for Data Mining Granular Model Definition Motivation and Examples Semantic Model Knowledge Systems and Granularity Data Mining as Generalization Data Mining Operators Data Mining Process Descriptive Model K-Descriptive Language K- Satisfaction and K- Truth Descriptive Language and Descriptive Model Granular Model Revisited: Satisfaction and Truth Future Directions Bibliography Granular Neural Networks Glossary Definition of the Subject Introduction Basic Architecture First Architecture of Granular Neural Network Layer 1: Linguistic Feature Extraction Layer Layer 2: Multi-FNNKD Layer Layer 3: Output Layer Second Architecture of Granular Neural Network Deep Granular Neural Networks Granular Learning Algorithms Local Forward-Wave Learning Procedure of Local Forward-Wave Learning Global Backward-Wave Learning Procedure of Global Backward-Wave Learning Forward-Wave-Backward-Wave Learning Procedure Genetic Forward-Wave-Backward-Wave Learning Applications in Bioinformatics Applications in Computational Web Intelligence Concepts of Input Data Overview of Implementation Performance Applications in Brain Informatics Conclusions Future Directions Bibliography Granulation of Knowledge: Similarity Based Approach in Information and Decision Systems Glossary Definition of the Subject and Its Importance Introduction: Basic Ideas Similarity Information and Decision Systems Granulation of Knowledge: General Ideas and Technique of Rough Inclusions A General Form of Similarity Granulation: Tools Granulation by Means of Rough Inclusions Mereological Theory of Concepts Rough Inclusions Calculus on Granules Some Principal Fields of Applications Reasoning About Uncertainty Reasoning in Distributed Systems Reasoning in Cognitive Schemes: Rough Neural Reasoning Applications in Data Mining: Granular Classifiers Granules from Parameterized Variants of Rough Inclusions μh in Classification of Data Complexity Issues Future Directions Bibliography Primary Literature Books and Reviews Multi-Granular Computing and Quotient Structure Glossary Definition of the Subject Introduction The Granularity Relation Truth Preserving Property Falsity Preserving Property Hierarchy The Computational Complexity Under Deterministic Model Combination The Construction of Quotient-Space Model The Combination of Quotient Spaces The Combination of Attribute Functions Future Directions Acknowledgments Bibliography Non-standard Analysis, an Invitation to Glossary Definition of the Subject Introduction Differential = Infinitesimal Difference? Structure of This Article Pre-Robinson Infinitesimals Recap: Naive Set Notations Ordered Field Extension Classical Augmented Infinities Non-Archimedean Infinitesimal Hyperreals The Construction of Ultimate-Reals Star-Extensions NS View of Continuous Functions NS Calculus NS Integral Calculus NS-ODE General Idea of NSA The Set-Terminology Naive Set Operations The Ultra-Power Construction Some Consequences of Transference Loeb Construction Measure Loeb Measure Anderson´s Wiener Process A Future for Non-standard Analysis? Bibliography Information System Design Using Fuzzy and Rough Set Theory Glossary Definition of the Subject and Its Importance Introduction Fuzzy Sets and Rough Sets Fuzzy Set Theory Rough Set Theory Rough Relational Database Information Theory Entropy and the Rough Relational Database Rough Fuzzy Relational Database Fuzzy Rough Relational Operators Difference Union Intersection Select Project Join Rough Set Modeling of Spatial Data Data Mining in Rough Databases Aggregation of Uncertain Information Future Directions Acknowledgments Bibliography Primary Literature Books and Reviews Rough Set Data Analysis Introduction Focusing Mechanism Definition of Rules Rough Sets Classification Accuracy and Coverage Probabilistic Rules Positive Rules Negative Rules Algorithms for Rule Induction Experimental Results Performance of Rules Obtained What Is Discovered? Interesting Rules Are Very Few Positive Rules in Differential Diagnosis of Headache Negative Rules in Differential Diagnosis of Headache Positive Rules in Meningitis Positive and Negative Rules in CVD Rule Discovery as Knowledge Acquisition and Decision Support Expert System: RH Evaluation of RH Discussion Hierarchical Rules for Decision Support Relations Between Rules Conclusions Bibliography Rule Induction, Missing Attribute Values, and Discretization Glossary Discretization LEM2 Algorithm LERS Machine Learning/Data Mining System Missing Attribute Values Rule Induction Definition of the Subject and Its Importance Introduction Discretization LEM2 Algorithm Procedure LEM2 Inconsistent Data Missing Attribute Values Lower and Upper Approximations Probabilistic Approximations MLEM2 Classification System Validation Future Directions Bibliography Social Networks and Granular Computing Glossary Definition of the Subject and Its Importance Introduction Mathematical Preliminaries Sets Multisets Relations Rough Set Theory: Functional Granulation Positional Analysis: Relational Granulation Modal Logic: The Bridge Propositional Multi-modal Logic Graded Modal Logic Hybrid Logic From Relational Granulation to Functional Granulation Structural Equivalences and Hybrid Logics Regular Equivalences and Multi-modal Logics Exact Equivalences and Graded Modal Logics Observational Equivalences Related Work Conclusion and Future Directions Bibliography Rough Sets in Decision-Making Entry Outline Glossary Definition of the Subject and Its Importance Introduction to Decision Support Using Rough Set Concept Classical Rough Set Approach to Classification Problems of Taxonomy Type Data Table and Indiscernibility Relation Approximations Dependence and Reduction of Attributes Decision Table and Decision Rules Explanation of the Classical Rough Set Approach by an Example Dominance-Based Rough Set Approach to Ordinal Classification Problems Dominance-Based Rough Set Approach (DRSA) Example Illustrating DRSA in the Context of Ordinal Classification Robust Ordinal Regression for DRSA DRSA on a Pairwise Comparison Table for Multiple Criteria Choice and Ranking Problems Multiple Criteria Choice and Ranking Problems The Pairwise Comparison Table Approximation by Means of Graded Dominance Relations Decision Rules Application of the Decision Rules and Final Recommendation Approximation by Means of Multi-graded Dominance Relations Dominance Without Degrees of Preference Example Illustrating DRSA in the Context of Multiple Criteria Choice and Ranking DRSA for Decision Under Uncertainty Basic Concepts Example Illustrating DRSA in the Context of Decision Under Uncertainty Multiple Criteria Decision Analysis Using Association Rules Interactive Multiobjective Optimization Using DRSA (IMO-DRSA) Example Illustrating IMO-DRSA in the Context of Multiobjective Optimization Characteristics of the IMO-DRSA Some Dedicated Applications of the IMO-DRSA: Financial Portfolio and Project Portfolio An Abstract Algebra for Reasoning About Multiple Criteria Decisions: The Bipolar Pawlak-Brouwer-Zadeh Lattice Conclusions Prospective Directions of Applications Bibliography Knowledge Engineering from Email Archives Glossary Definition of the Subject Introduction Email Format History Email System Email Security and SPAM Prevention Email Clients Email Archiving Compliance/Surveillance eDiscovery Archiving Modes Archiving Tasks Mail Parsing Storing of Mails User Association Indexing and Querying Implicit/Hidden Knowledge in Email Archives Future Directions Bibliography Fuzzy Similarity for Parallel Function Computation Model Background OpenCL (Open Computing Language) Fuzzy Similarity for Natural Language Processing (NLP) Introduction word2vec Mathematical Transformations Book Cataloging Translations in Natural Language Processing Bibliography Mereology Glossary Introduction First-Order Logic First-Order Mereological Axioms and Theories Alternative Formulations Models of Mereological Theories Decidability Concerning First-Order Mereological Theories Fuzzification of Mereology Future Directions Acknowledgments Bibliography Primary Literature Books and Reviews Variable Precision Approximations of Rough Sets Introduction Basics of Pawlak´s Rough Sets Approximation Space Set Approximation Regions Variable Precision Rough Set Model Set Frequency and Conditional Frequency Approximation Regions in the VPRSM Absolute Set Approximation Regions Decision Tables Learned from Data Attributes Deterministic and Nondeterministic Decision Tables Probabilistic Decision Tables Dependencies in Probabilistic Decision Tables γ - Dependency Measure λ - Dependency Measure Reduction and Significance of Attributes Attribute Reduct Significance of Attributes Final Remarks Bibliography Algebraic Models and Granular Computing Glossary Definition of the Subject Introduction: Algebraic Models Example: Boolean Algebras and Classical Logic Example: Some Non-Boolean Algebras and Nonclassical Logics E1. Positive Implication Algebras Equational Definability of PIA Positive Implication Algebra of Sets E2. Implication Algebras Implication Algebra IA Equational Definability of IA Implication Algebra of Sets E3. Peirce Algebras Peirce Algebra PA Equational Definability of PA Peirce Algebra of Sets Filters in Peirce Algebras E4. Quasi-Boolean Algebras and De Morgan Lattices Quasi-Boolean Algebra De Morgan Lattice QBA of Sets Granular Computing S5 Rough Algebra Construction of Modal Lindenbaum-Tarski Algebra Modal Lindenbaum-Tarski Algebra S5 Rough Algebra Construction Properties of the S5 Rough Algebra Questions and Observations Topological Rough Algebras Generalized Rough Sets, L-Fuzzy, and LT-Fuzzy Sets LT-Fuzzy Sets Generalized Rough Sets Approximation Space Topological Space Approximation Topological Space Topological and Rough Equalities Rough, L-Fuzzy, and LT-Fuzzy Sets Bibliography Set-Valued Mapping for Generalized Rough Approximation and Application to Clustering Algorithm Introduction Rough Approximation and Generalizations A Class of Generalizations A Set-Valued Mapping and Rough Approximations The Case of Neighborhood Soft Boundary in an Euclidean Space Application to K-Means Clustering K-Means Algorithm Allocation Rule Rough Approximations Optimization Problem An Illustrative Example Conclusion Bibliography Fuzzy System Representations of Stochastic Systems Glossary Definition of the Subject Introduction Background of Stochastic Systems Fuzzy Reasoning Meaning of Continuous Stochastic Systems Fuzzy Reasoning Representations of Double-Inputs and Single-Output Continuous Stochastic Systems Fuzzy Reasoning Representations of Discrete Stochastic Systems Reducibility in the Transformations Between Fuzzy Systems and Stochastic Systems The Uncertainty Systems with One-Dimensional Random Variable and Their Representations Typical Case 1: Pure Certainty Systems Typical Case 2: Pure Stochastic Systems Typical Case 1* Typical Case 2* Unification on Uncertainty Systems Conclusions Bibliography Function Approximation Property of Fuzzy Systems and Its Error Analysis Glossary Definition of the Subject Introduction Structures of Fuzzy Systems Function Approximation Property of Fuzzy Systems The Approximate Remainder Formula of fn(x) to Continuous Function s(X) The Error Estimation Between Fuzzy System and fn(x) Simulation Example Conclusions Bibliography Issues in Access Control and Privacy for Big Data Glossary Definition of the Subject Introduction Access Control and Privacy Access Control Privacy Access Control Versus Privacy 4 Vs of Big Data Volume Variety Velocity Veracity Putting it All Together Summary and Future Directions Bibliography Security Situational Awareness: A Practical Approach for a Complex Heterogeneous Environment Glossary Definition of the Subject Introduction Security-Relevant Data That Is Useful for Protecting a Data Center Vulnerability Data Network Flow Data DNS Data Intrusion Detection System Data Centralized Log Data Risk-Partitioning the Mountain of Data Partitioning Based on Acceptable Risk Partitioning Based on High Risk Partitioning Based on Unknown Risk Results of Partitioning Data Enrichment to Facilitate Actionable Security Event Detection Detecting Actionable Security Events Vulnerabilities System´s Software Vulnerability Vulnerabilities Due to Misconfiguration of Computer Systems or Networks Vulnerability-Related Policy Violation Intrusions Intrusion Detection System Events Brute-Force Attacks Suspicious Logins APTs, Malware Beacons, and Back Doors Suspicious DNS Activity Policy Violations Insider Threats and Data Exfiltration Future Directions Bibliography Security with Privacy Introduction Privacy Enhancing Techniques for Security Privacy-Preserving Data Matching Privacy-Preserving Collaborative Data Mining Privacy-Preserving Biometrics Authentication Privacy-Preserving Fine-Grained Access Control for Cloud Data Future Directions Concluding Remarks Acknowledgments Glossary References Privacy Preservation in Big Data Analytics Definition of Subject Glossary Introduction What Is Big Data Open Data for Governments Big Data: Issues and Challenges Privacy Law Privacy Preservation Data Publishing Well-Known Examples for Privacy Preservation in Re-identification Problems Techniques for Privacy Preserving Data Publishing k-Anonymity on Relational Data l-Diversity and t-Closeness on Relational Data Anonymization Methods on Transactional Data Anonymization Methods on Social Network Data Privacy Versus Utility Differential Privacy The Challenges on Big Data Privacy Preservation Current Techniques for Privacy Preservation on Big Data Opening Issues to Privacy Preservation on Big Data Privacy Preservation Big Data Publishing Issues Privacy Preservation Big Data Mining Issues Summary Bibliography Access Control for XML Big Data Applications Introduction Use-Case Background UML Extensions for Document Modeling Extensions for Document Security Policy Modeling and Integration Integrating Local Security Policies into a Global Security Policy Assumptions and Equivalence Finding Integration Process for Local Security Policy Sets Resolving Conflicts of Integrated Security Rule Sets Creating the Global Security Policy Related Work Conclusion Bibliography A Theory of Approximate Arithmetic for Numerical Analysis: A Mathematical Foundation Glossary Definition of the Subject Introduction SRNC: Approximation for Numerical Computing SRNC: A Paradox? In Approximate Addition Backgrounds: Lemmas, Notations, and Conventions Approximations, Metersticks, and Topologies Measuring Functions and Topologies Approximate Arithmetic Semi-group Semi-group Topological Fields Isomorphisms of Ordered Fields Conclusions References Interval Computing Definition of Interval Computing Introduction Motivation Early History Modern History Key Information Basic Definitions and Properties Current Uses Reference Resources General and Introductory Books Applications Software Other Resources Future Directions References Aggregation Operators and Soft Computing Glossary Definition of the Subject Introduction Applications Aggregation, Integration, and Fusion Aggregation Operators and Their Construction Some Aggregation Operators Quasi-Arithmetic Means Weighted Means and Quasi-Weighted Means Losonczi´s Means Linear Combination of Order Statistics and OWA Operators The WOWA Operator Fuzzy Integrals Future Directions Bibliography Primary Literature Books and Reviews Evolving Fuzzy Systems Glossary Definition of the Subject Introduction Evolving Clustering Evolving TS Fuzzy Systems Evolving Fuzzy Classifiers Evolving Fuzzy Controllers Application Case Studies Self-Calibrating Intelligent Sensors for Process Industries Adaptive Real-Time Classifiers (Image, Land-Mark, Robotic) Predictive Models (Air-Conditioning, Financial Time-Series, Benchmark Data Sets) Fault Detection and Prognostics Speech Signal Reconstruction Future Directions Acknowledgments Bibliography Primary Literature Books and Reviews Fuzzy Logic, Type-2 and Uncertainty Glossary Definition of the Subject Introduction Imprecision Vagueness or Linguistic Uncertainty Type-2 Fuzzy Systems Union of Type-2 Fuzzy Sets Intersection of Type-2 Fuzzy Sets Generalized Type-2 Fuzzy Systems Interval Type-2 Fuzzy Sets and Systems Interval Type-2 Fuzzy Sets Interval Type-2 Fuzzy Logic Systems Applications Future Directions Bibliography Fuzzy Optimization Glossary Definition of the Subject Introduction Basics of Fuzzy Set Theory Extension Principles Basics of Possibility Theory Semantics of Fuzzy Sets and Possibility Distributions in Fuzzy Optimization Fuzzy Relations Basics of Fuzzy Sets and Possibility Theory in Optimization Classical Approaches to Fuzzy Optimization Fuzzy Programming Possibilistic Programming Programming with Fuzzy and Possibilistic Components Possibilistic, Interval, Cloud, and Probabilistic Optimization Utilizing IVPM Future Directions Bibliography Foundations of Fuzzy Sets Theory Glossary Definition of the Subject Introduction Fuzzy Sets: Basic Definition and Properties The Idea of a Fuzzy Set Some Extensions of the Concept of Zadeh´s Fuzzy Set Basic Definition and Properties Related to Fuzzy Sets Basic Operations on Fuzzy Sets Fuzzy Relations Linguistic Variable, Fuzzy Conditional Statement, and Compositional Rule of Inference The Extension Principle Fuzzy Numbers Fuzzy Events and Their Probabilities Defuzzification of Fuzzy Sets Fuzzy Logic: Basic Issues Bellman and Zadeh´s General Approach to Decision-Making Under Fuzziness Multistage Decision-Making (Control) Under Fuzziness Concluding Remarks Bibliography Hybrid Soft Computing Models for Systems Modeling and Control Glossary Definition of the Subject Introduction Genetic Algorithm for Optimization Genetic Algorithm for Optimization Evolution of Fuzzy Systems Application to Anesthesia Control Anesthesia Control Using Fuzzy Logic Characteristics of the Fuzzy Controller Genetic Algorithm Specification Representation of the Chromosome Simulation Results for the Case of Anesthesia Control Application to the Control of the Bar and Ball System Simulation Results with the Complete HGA Simulation Results with a Method Dased Only on Mutation Simulation Results with a Method Based Only on Crossover Comparison of the Simulation Results Hierarchical Genetic Algorithms for Neural Networks Experimental Results for Time Series Prediction Prediction with a Neural Network Prediction with an Adaptive Neuro-Fuzzy Inference System (ANFIS) Prediction with an Interval Type-2 TSK Fuzzy Model Prediction with a Neuro-Genetic Model Prediction with Type-1 Fuzzy Models Optimized with Genetic Algorithms Conclusions Future Directions Bibliography Neuro-fuzzy Systems Glossary Definition of the Subject Introduction Fuzzy Reasoning Logical Approach to Fuzzy Inference Reasoning Mamdani Approach to Fuzzy Inference Reasoning Description of Fuzzy Inference Systems Logical-Type Neuro-fuzzy Systems Mamdani-Type Neuro-fuzzy Systems Simplified Neuro-fuzzy Systems Takagi-SugenoNeuro-fuzzy Systems Neuro-fuzzy Systems with Weights Neuro-fuzzy Systems for Pattern Classification Learning Criteria Isolines Method Future Directions Acknowledgments Bibliography Possibility Theory Glossary Definition of the Subject and Its Importance Introduction Historical Background G. L. S. Shackle D. Lewis L. J. Cohen L. A. Zadeh Basic Notions of Possibility Theory Qualitative Possibility Theory Nonmonotonic Inference Possibilistic Logic Decision-Theoretic Foundations Quantitative Possibility Theory Possibility as Upper Probability Conditioning Probability-Possibility Transformations From Possibility to Probability From Objective Probability to Possibility Possibility Distributions Induced by Prediction Intervals Subjective Possibility Distributions Mean Values in Possibility Theory Applications and Future Directions Bibliography Rough Sets: Foundations and Perspectives Glossary Definition of the Subject Introduction Approximation Spaces Framework for an Approximation Space Rough Sets Information Tables Decision Systems and Decision Rules Dimensionality Reduction Summary Future Directions Acknowledgments Bibliography Primary Literature Further Readings - Books Further Readings - Transactions on Rough Sets Further Readings - Special Issues of Journals Further Readings - Proceedings of International Conferences Introduction to Soft Computing Statistics with Imprecise Data Glossary Definition of the Subject Introduction Mathematical Modeling of Imprecise Data Fuzzy Random Variables Statistical Analysis of Fuzzy Data Corresponding to Fuzzy Perceptions of Existing Real-Valued Data Fuzzy Estimation Fuzzy Statistical Tests Statistical Analysis of Existing Fuzzy-Valued Data Fuzzy Estimation Statistical Tests Future Directions Bibliography Primary Literature Books and Reviews Some Remarks on the Foundations of Numerical Analysis Glossary Definition of Numerical Analysis Introduction Early History Modern History Cn Modern Numerical Analysis Classical References Future Directions References Index