Joan Torrens

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.

A new linguistic computational model based on discrete fuzzy numbers for computing with words

In recent years, several different linguistic computational models for dealing with linguistic information in processes of computing with words have been proposed. However, until now all of them rely on the special semantics of the linguistic terms, usually fuzzy numbers in the unit interval, and the linguistic aggregation operators are based on aggregation operators in [0,1]. In this paper, a linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers is presented ensuring the accuracy and consistency of the model. In this framework, no underlying membership functions are needed and several aggregation operators defined on the set of all discrete fuzzy numbers are presented. These aggregation operators are constructed from aggregation operators defined on a finite chain in accordance with the granularity of the linguistic term set. Finally, an example of a multi-expert decision-making problem in a hierarchical multi-granular linguistic context is given to illustrate the applicability of the proposed method and its advantages.

On a class of operators for expert systems

We use the concept of directed algebra (closely related to De Morgan triplets) to modelize connectives in expert systems when linguistic terms are introduced. Mainly this article describes all directed algebra structures on a totally ordered finite set.

On locally internal monotonic operations

This paper deals with monotonic binary operations F: [0, 1]2 → [0, 1] with the property (called locally internal property) that the value at any point (x, y) is always one of its arguments x, y. After stating a theorem that characterizes this kind of operations, some special cases are studied in detail by considering additional properties of the operation: commutativity, existence of a neutral element and associativity. In case of locally internal, associative monotonic operations with neutral element, a characterization theorem gives an improvement of a well-known theorem of Czogala and Drewniak on idempotent, associative and increasing operations with neutral element, as well as an improvement of a characterization theorem for left (and right) continuous, idempotent uninorms.

7 – Triangular norms on discrete settings

Triangular norms and conorms, as well as copulas acting on finite or infinite discrete chains are studied. Special attention is paid to the divisibility of these operations.

The distributivity condition for uninorms and t-operators

In this work we study the functional equation given by the distributivity condition: F(x, G(y,z)) = G(F(x, y),F(x,z)) for all x,y,z ∈ [0,1], where the unknown functions F,G are uninorms and/or t-operators. We characterize all the solutions in the four possible cases: (i) when F,G are t-operators, (ii) when F is a t-operator and G a uninorm, (iii) when F is a uninorm and G a t-operator and, finally, (iv) when F, G are uninorms.

Distributivity and conditional distributivity of a uninorm and a continuous t-conorm

The open problem recalled by Klement in the Linz2000 closing session, related to distributivity and conditional distributivity of a uninorm and a continuous t-conorm, is solved for the most usual known classes of uninorms. From the obtained results, it is deduced that distributivity and conditional distributivity are equivalent for these cases. It is remarkable that solutions appear involving not only strict t-conorms but also ordinal sums of the maximum with a strict t-conorm. Conversely, the distributivity of a t-conorm over a uninorm is also studied leading only to already known solutions. Moreover, the dual case of distributivity and conditional distributivity involving uninorms and continuous t-norms is also solved, proving again the equivalence of both kinds of distributivities

t‐Operators and uninorms on a finite totally ordered set

This paper presents a detailed study of two classes of operators on a finite totally ordered set of labels L: t‐operators and uninorms. Both kinds of operators (on [0, 1]) are introduced as generalizations of t‐norms and t‐conorms. We characterize these operators on L as special combinations of operators of directed algebras in a similar way as they are characterized in the case of [0, 1] as special combinations of t‐norms and t‐conorms. We also study duality of these operators with respect to the only negation N on L, and we give the number of different t‐operators and uninorms that exist on L, related to the number of elements in L. ©1999 John Wiley & Sons, Inc.

Some remarks on the characterization of idempotent uninorms

In this paper the characterization of idempotent uninorms given in [21] is revisited and some technical aspects are corrected. Examples clarifying the situation are given and the same characterization is translated in terms of symmetrical functions. The particular cases of left-continuity and right-continuity are studied retrieving the results in [7].

On a new class of fuzzy implications: h-Implications and generalizations

A new class of fuzzy implications called the h-implications is introduced. They are implications generated from an additive generator of a representable uninorm in a similar way of Yagers f- and g-implications which are generated from additive generators of continuous Archimedean t-norms and t-conorms. Basic properties of these implications are studied in detail. Modifications and generalizations of the initial definition are presented and their properties studied and compared between them. One of the modifications, called (h,e)-implications, is another example of a fuzzy implication satisfying the exchange principle but not the law of importation for any t-norm, in fact for any function F : [0,1]^2->[0,1].