Sebastia Massanet

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 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].

The law of importation versus the exchange principle on fuzzy implications

Some open problems on fuzzy implications dealing with the so-called importation law are studied and totally or partially solved in this work. A weaker version of the law of importation, called the weak law of importation, is introduced. The relationships of these two properties and the exchange principle are studied. In particular, it is proved that the law of importation is stronger than the exchange principle. On the other hand, the three properties are equivalent for some kind of fuzzy implications, those that satisfy a boundary property. Along this study, new characterizations of (S,N)-implications, R-implications and their counterparts for uninorms based on the weak law of importation are showed.

A survey on the existing classes of uninorms

This paper wants to be a compilation of the different existing classes of uninorms. From their introduction, uninorms have been extensively studied not only as aggregation operators but also as logical connectives. The study of both aspects has produced many results on this kind of operators and many different classes have appeared. This work does not aim to be exhaustive since it is not possible to recall all known results about all classes of uninorms in a reduced space. Thus, we only want to state the general research lines of these classes in the two main frameworks where uninorms have been studied: the unit interval (0, 1) and the discrete setting. However, we will also compile the references where more details about all the existing classes of uninorms can be found, for the convenience of the interested reader. Uninorms in other frameworks are also recalled and finally, a section devoted to applications of uninorms is included.

On the characterization of Yager's implications

In this paper, characterizations of Yagers f- and g-implications are presented. Since their introduction in 2004, the properties of these implications have been studied in detail but they have not been characterized yet. The characterizations given here are based on the law of importation, a functional equation that has been extensively studied. Moreover, particular characterizations of Reichenbach, Yager and Goguen implications are derived from these results.

Fuzzy Implications: Past, Present, and Future

Fuzzy implications are a generalization of the classical two-valued implication to the multi-valued setting. They play a very important role both in the theory and applications, as can be seen from their use in, among others, multivalued mathematical logic, approximate reasoning, fuzzy control, image processing, and data analysis. The goal of this chapter is to present the evolution of fuzzy implications from their beginnings to the current days. From the theoretical point of view, we present the basic facts, as well as the main topics and lines of research around fuzzy implications. We also devote a specific section to state and recall a list of main application fields where fuzzy implications are employed, as well as another one to the main open problems on the topic.

Some interesting properties of the fuzzy linguistic model based on discrete fuzzy numbers to manage hesitant fuzzy linguistic information

Graphical abstractDisplay Omitted HighlightsProperties of the fuzzy linguistic model based on discrete fuzzy numbers are analysed.This model is used to handle hesitant fuzzy linguistic information.This model includes the hesitant fuzzy linguistic term sets model.Some advantages of the model based on discrete fuzzy numbers are pointed out.A fuzzy decision making model based on discrete fuzzy numbers is proposed. The management of hesitant fuzzy information is a topic of special interest in fuzzy decision making. In this paper, we focus on the use and properties of the fuzzy linguistic modelling based on discrete fuzzy numbers to manage hesitant fuzzy linguistic information. Among these properties, we can highlight the existence of aggregation functions with no need of transformations or the possibility of a greater flexibilization of the opinions of the experts, even using different linguistic chains (multigranularity). Furthermore, based on these properties we perform a comparison between this model and the one based on hesitant fuzzy linguistic term sets, showing the advantages of the former with respect to the latter. Finally, a fuzzy decision making model based on discrete fuzzy numbers is proposed.

An Overview of Construction Methods of Fuzzy Implications

Fuzzy implications are useful in a wide range of applications. For these practical purposes, different classes of implications are used. Depending on the concrete application the implication is going to perform, several additional properties have to be fulfilled. In this paper, we recall briefly the most used classes of implications, (S,N) and R-implications, QL and D-operations, their generalizations to other aggregation functions and Yager’s implications, and we show the new construction methods presented in recent years. These construction methods vary from implications defined from general aggregation functions or fuzzy negations, to implications generated from one or two initial implications. For every single class, we determine which additional properties are satisfied.

On the Choice of the Pair Conjunction–Implication Into the Fuzzy Morphological Edge Detector

In this paper, the fuzzy morphological gradients from the fuzzy mathematical morphologies based on t-norms and conjunctive uninorms are deeply analyzed in order to establish which pair of conjunction and fuzzy implications are optimal, in accordance with their performance in edge detection applications. A novel three-step algorithm based on the fuzzy morphology is proposed. The comparison is performed by means of the so-called Pratts figure of merit. In addition, a statistical analysis is carried out to study the relationship between the different configurations and to establish a classification of the conjunctions and implications considered. Both the objective measure and the statistical analysis conclude that the pairs nilpotent minimum t-norm and the Kleene-Dienes implication, and the idempotent uninorm obtained with the classical negation as a generator and its residual implication, are the best configurations in this approach, because they also obtain competitive results with respect to other approaches.

Threshold generation method of construction of a new implication from two given ones

In this paper, a new construction method of a fuzzy implication from two given ones, called threshold generation method, is introduced. It is a generalization of the way of construction of the recently introduced h-implications, which are fully characterized in this work. The threshold generation method allows to control, up to a certain level, the increasingness on the second variable of the fuzzy implication through an adequate scaling on that variable of the two given implications. The natural propagation of the most usual properties of fuzzy implications from the initial ones to the constructed implication is studied and the necessary and sufficient conditions in order to ensure this propagation are presented. In particular, the preservation of the contrapositive symmetry on threshold generated implications needs of another construction method of a fuzzy implication from a given one and a fuzzy negation.