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

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