Explainable Neural Networks Based on Fuzzy Logic and Multi-criteria Decision Tools

Description

The research presented in this book shows how combining deep neural networks with a special class of fuzzy logical rules and multi-criteria decision tools can make deep neural networks more interpretable – and even, in many cases, more efficient. Fuzzy logic together with multi-criteria decision-making tools provides very powerful tools for modeling human thinking. Based on their common theoretical basis, we propose a consistent framework for modeling human thinking by using the tools of all three fields: fuzzy logic, multi-criteria decision-making, and deep learning to help reduce the black-box nature of neural models; a challenge that is of vital importance to the whole research community. Foreword Preface Introduction—Aggregation and Intelligent Decision Contents List of Figures List of Tables Elements of Nilpotent Fuzzy Logic 1 Connectives: Conjunctions, Disjunctions and Negations 1.1 Introduction 1.2 Preliminaries 1.2.1 Negations 1.2.2 Triangular Norms and Conorms 1.3 Characterization of Strict Negation Operators 1.4 Nilpotent Connective Systems 1.4.1 Structural Properties of Connective Systems 1.4.2 Consistent Connective Systems 1.5 Summary References 2 Implications 2.1 Introduction 2.2 Preliminaries 2.3 R-Implications in Bounded Systems 2.4 S-Implications in Bounded Systems 2.4.1 Properties of iSn, iSd and iSc 2.4.2 S-Implications and the Ordering Property 2.5 A Comparison of Implications in Bounded Systems 2.6 Min and Max Operators in Nilpotent Connective Systems 2.7 Summary References 3 Equivalences 3.1 Introduction 3.2 Preliminaries 3.3 Equivalences in Bounded Systems 3.3.1 Properties of ec(x,y) and ed(x,y) 3.4 Dual Equivalences 3.4.1 Properties of bared and barec 3.5 Arithmetic Mean Operators in Bounded Systems 3.6 Aggregated Equivalences 3.6.1 Properties of the Aggregated Equivalence Operator 3.7 Applications 3.8 Summary References 4 Modifiers and Membership Functions in Fuzzy Sets 4.1 Introduction 4.2 Unary Operators in Nilpotent Logical Systems 4.2.1 Possibility and Necessity as Unary Operators Derived from Multivariable Operators 4.2.2 Drastic Unary Operators 4.2.3 Composition Rules 4.2.4 Multivariable Operators Derived from Unary Operators 4.2.5 A General Framework: The α, β, γ- Model 4.3 Unary Operators Induced by Negation Operators 4.4 Membership Functions 4.5 Non-membership Functions 4.6 Summary References Decision Operators 5 Aggregative Operators 5.1 Introduction 5.2 Preliminaries 5.3 Shifting Transformations on the Generator Functions – A General Parametric Formula 5.4 The Weighted General Operator 5.5 Properties of the General and the Weighted General Operator 5.5.1 The De Morgan Property 5.5.2 Bisymmetry 5.6 The Two-Variable General and Weighted Aggregative Operator 5.7 Summary References 6 Preference Operators 6.1 Introduction 6.2 Operators of Nilpotent Systems - A General Framework 6.2.1 Normalization of the Generator Functions 6.2.2 The General Parametric Operator 6.2.3 The Unary Operators: Negation, Modifiers and Hedges 6.3 Preference Modeling 6.4 Properties of the Preference Operator 6.4.1 Basic Properties 6.4.2 Ordering Properties 6.4.3 Preference and Negation 6.4.4 Preference, Conjunction and Disjunction 6.4.5 Preference and Aggregation 6.4.6 Additive Transitivity 6.4.7 Bisymmetry and the Common Base Property 6.4.8 Preference and Unary Operators 6.5 Summary References Learning and Neural Networks 7 Squashing Functions 7.1 Introduction 7.2 Łukasiewicz Operators 7.3 Approximation of the Cutting Function 7.3.1 The Sigmoid Function 7.3.2 The Interval [a,b] Squashing Function 7.3.3 The Error of the Approximation 7.4 Approximation of Piecewise Linear Membership Functions 7.5 Summary References 8 Learning Rules 8.1 Introduction 8.2 Problem Definition and Solution Outline 8.3 Preliminaries 8.4 The Structure and Representation of the Rules 8.5 The Optimization Process 8.5.1 Rule Optimization by GA 8.5.2 A Gradient-Based Local Optimization of Memberships 8.6 Applications 8.7 Summary References 9 Interpretable Neural Networks Based on Continuous-Valued Logic and Multi-criteria Decision Operators 9.1 Introduction 9.2 Related Work 9.3 Nilpotent Logical Systems and Multicriteria Decision Tools 9.4 Nilpotent Logic-Based Interpretation of Neural Networks 9.5 Playground Examples 9.5.1 XOR 9.5.2 Preference 9.6 Summary References 10 Conclusions