Miquel Monserrat

A Survey on Fuzzy Implication Functions

One of the key operations in fuzzy logic and approximate reasoning is the fuzzy implication, which is usually performed by a binary operator, called an implication function or, simply, an implication. Many fuzzy rule based systems do their inference processes through these operators that also take charge of the propagation of uncertainty in fuzzy reasonings. Moreover, they have proved to be useful also in other fields like composition of fuzzy relations, fuzzy relational equations, fuzzy mathematical morphology, and image processing. This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators. The four most usual ways to define these implications are recalled and their characteristic properties stated, not only in the case of [0,1] but also in the discrete case.

On the representation of fuzzy rules

In fuzzy logic, connectives have a meaning that, can frequently be known through the use of these connectives in a given context. This implies that there is not a universal-class for each type of connective, and because of that several continuous t-norms, continuous t-conorms and strong negations, are employed to represent, respectively, the and, the or, and the not. The same happens with the case of the connective If/then for which there is a multiplicity of models called T-conditionals or implications. To reinforce that there is not a universal-class for this connective, four very simple classical laws translated into fuzzy logic are studied.

On left and right uninorms

This paper presents a characterization of a new family of operators. Namely, all non-decreasing, associative binary operators U: [0, 1]2 → [0, 1] with a left (or right) neutral element e ∈ [0, 1], and such that they satisfy an additional hypothesis on continuity which is called here left (or right) pseudocontinuity.

On left and right uninorms on a finite chain

The main concern of this paper is to introduce and characterize the class of operators on a finite chain L, having the same properties of pseudosmooth uninorms but without commutativity. Moreover, in this case it will only be required the existence of a one-side neutral element. These operators are characterized as combinations of AND and OR operators of directed algebras (smooth t-norms and smooth t-conorms) and the case of pseudosmooth uninorms is retrieved for the commutative case.

A characterization of (U,N), RU, QL and D-implications derived from uninorms satisfying the law of importation

The law of importation is exhaustively studied for fuzzy implications derived from uninorms. All widely used classes of implications, that is (U,N), RU, QL and D-implications derived from most usual classes of uninorms are investigated. Along this study many new solutions of the law of importation appear different from those already known for implications derived from t-norms and t-conorms.

Aggregation operators with annihilator

This paper is devoted to the study of a special kind of aggregation operators: commutative, non-decreasing binary operators F on [0,1] with annihilator and such that and . A characterization of this kind of operators is given, including many examples and properties in the general case. Special attention is paid to the associative case, leading to a characterization by means of a median expression. This type of operators can be viewed as a generalization of both uninorms and nullnorms.

Migrative uninorms and nullnorms over t-norms and t-conorms

In this paper the notions of α-migrative uninorms and nullnorms over a fixed t-norm T and over a fixed t-conorm S are introduced and studied. All cases when the uninorm U lies in any one of the most usual classes of uninorms are analyzed, characterizing all solutions of the migrativity equation for all possible combinations of U and T and for all possible combinations of U and S. A similar study is done for nullnorms.

Kernel aggregation functions on finite scales. Constructions from their marginals

Abstract The study of discrete aggregation functions (those defined on a finite chain) with some kind of smoothness has been extensively developed in last years. Many different kinds of aggregation functions have been characterized in this context. In this paper discrete aggregation functions with the kernel property (which implies the smoothness property) are investigated. Some properties and characterizations, as well as some construction methods for this kind of discrete aggregation functions are studied. It is also investigated when the marginal functions of a discrete kernel aggregation function fully determine it.

On two types of discrete implications

This paper deals with two kinds of implications defined from t-norms, t-conorms and strong negations on a finite chain L: those defined through the expressions I(x,y)=S(N(x),T(x,y)) and I(x,y)=S(T(N(x),N(y)),y). They are called QL-implications and NQL-implications respectively. We mainly study those QL- and NQL-implications derived from smooth t-norms and smooth t-conorms. It is characterized when functions defined in these ways are implication functions, and their analytical expressions are given. It is proved that both kinds of implications agree. Some additional properties are studied like contrapositive symmetry, the exchange principle and others. In particular, it is proved that contrapositive symmetry holds if and only if S is the only Archimedean t-conorm on L, and T jointly with its N-dual t-conorm satisfy the Frank equation. Finally, some QL- and NQL-implications are also derived from non-smooth t-norms or non-smooth t-conorms and many examples are given showing that in this non-smooth case, QL- and NQL-implications remain strongly connected.

Algebraic transformation of unary partial algebras II: single-pushout approach

The transformation of total graph structures has been studied from the algebraic point of view over more than two decades now, and it has motivated the development of the so-called double-pushout and single-pushout approaches to graph transformation. In this article we extend the double-pushout approach to the algebraic transformation of partial many-sorted unary algebras. Such a generalization has been motivated by the need to model the transformation of structures which are richer and more complex than acyclic graphs and hypergraphs. The main result presented in this article is an algebraic characterization of the double-pushout transformation in the categories of all homomorphisms and all closed homomorphisms of unary partial algebras over a given signature, together with a corresponding operational characterization which may serve as a basis for implementation. Moreover, both categories are shown to satisfy the strongest of the HLR (High Level Replacement) conditions with respect to closed monomorphisms. HLR conditions are fundamental to rewriting because they guarantee the satisfaction of many rewriting theorems concerning confluence, parallelism and concurrency.