Adolfo R. de Soto

On antonym and negate in fuzzy logic

The abilities to speak well and to conceptualize seem to be closely linked. It has been maintained that the human brain has a preference for binary oppositions or polarities. The notions of antonym and negate are examples of polarity between the pairs of predicates P−no P, P−ant P. Other characteristics as mutual exclusivity, complementation, or reciprocity are applied, in some cases, to them. However, if negation is a general phenomenon in natural languages, the use of antonyms is more usual for graduate predicates. For this reason, antonyms were considered very early in fuzzy set theory. 1–4 In this work, some relations between antonyms and negates are analyzed in the frame of fuzzy set theory, showing both similarities and dissimilarities between these two concepts. The last goal is to get automatic methods to build concepts with an adequate and easy interpretation. This paper is an experimental‐theoretic intent on the way of establishing a mathematical model of antonymy in agreement both with some linguistic facts and with some uses in fuzzy control. ©1999 John Wiley & Sons, Inc.

Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator

This work shows a method to obtain a hierarchy of partitions on the universe [0,1] in such a way that each of them is compatible with a refinement of Lukasiewicz indistinguishability operator. The classes of the partition at a given level present a relation of antonymy between them. Moreover, the partition at a certain level can be seen as the refinement of a previous level by means of a class of linguistic modifiers. Due to this fact, they seem appropriate for modeling linguistic labels of a linguistic variable. The associated indistinguishability operators show the increasing granularity when the number of classes rises up.

A hierarchical model of a linguistic variable

In this work a theoretical hierarchical model of dichotomous linguistic variables is presented. The model incorporates certain features of the approximate reasoning approach and others of the Fuzzy Control approach to Fuzzy Linguistic Variables. It allows sharing of the same hierarchical structure between the syntactic definition of a linguistic variable and its semantic definition given by fuzzy sets. This fact makes it easier to build symbolic operations between linguistic terms with a better grounded semantic interpretation. Moreover, the family of fuzzy sets which gives the semantics of each linguistic term constitutes a multiresolution system, and thanks to that any fuzzy set can be represented in terms of the set of linguistic terms. The model can also be considered a general framework for building more interpretable fuzzy linguistic variables with a high capacity of accuracy, which could be used to build more interpretable Fuzzy Rule Based Systems (FRBS).

Multi-dimensional aggregation of fuzzy numbers through the extension principle

In this paper we propose the problem of obtaining a procedure to aggregate fuzzy numbers in such a way that the output is also a fuzzy number. To do this, we use the Zadehs Extension Principle applied to multi-dimensional numerical functions which satisfy certain conditions, obtaining multi-dimensional aggregation functions on the lattice of fuzzy numbers. Special attention is given to the case of trapezoidal fuzzy numbers.

Hierarchical Linguistic Variables

The linguistic terms of a Linguistic Variable classify the numeric values of their Universe. In that sense, they are categories which are defined over a space of values. In this work a hierarchical model of a linguistic variable is analyzed from the point of view of the principles of categorization and hierarchy theory. It is shown how several partial orders can be defined over the hierarchical model in correspondence with two different interpretations of the hierarchical structure of the linguistic variable. Several methods to obtain the fuzzy set which gives the semantic value of a linguistic term in this family of linguistic variables are studied.

Some sets of indistinguishability operators as multiresolution families

Multiresolution is a general mathematical concept that allows us to study a property by means of several changes of resolution. From a fixed resolution, a coarser projection can be calculated and then the changes between a finer resolution and a coarser one can be studied. That information can give a good knowledge about the problem under consideration. Also using multiresolution techniques it is possible to present information with a higher or a lower detail, given a way to get the adequate granularity or abstraction for a context.The granularity of a system can be obtained or modeled by the use of indistinguishability operators. In this work the relation between indistinguishability operators and multiresolution theory is studied and several methods to build families of indistinguishability operators with multiresolution capacities are given.

Uniform Fuzzy Partitions with Cardinal Splines and Wavelets: Getting Interpretable Linguistic Fuzzy Models

In an abridged form, two main steps are habitual to build a Fuzzy Rule Model from a data set: 1. To select the shape and distribution of the linguistic labels set for each variable 2. To tune the set of rules to the training data set.

Some Comments on Ordinary Reasoning with Fuzzy Sets

The main goal of Computing with Words is essentially a calculation allowing to automate a part of the reasoning done thanks to the natural language. Fuzzy Logic is the main tool to perform this calculation because it is be able to represent the most common kind of predicates in natural language, graded predicates, in terms of functions, and to calculate with them. However there is still not an adequate framework to perform this task, commonly referred to as commonsense reasoning. This chapter proposes a general framework to model a part of this type of reasoning. The fundamental fact of this framework is its ability to adequately represent noncontradiction, the minimum condition for considering a reasoning as valid. Initially, the characteristics of the commonsense reasoning are analyzed, and a model for the crisp case is shown. After that the more general case in which graded predicates are taken under consideration is studied.

On Linguistic Variables and Sparse Representations

Linguistic variables can be seen as dictionaries to represent data. In fields as Signal Processing or Machine Learning is usual to use or to search redundant dictionaries to promote sparse representations. This kind of representations present several interesting properties as a high generalization capacity, simplification and economy, among others. In this work, a revision of the main methods to obtain sparse representations and their possible application to model with linguistic variables and Fuzzy Rule Systems is done.

Classes of fuzzy sets with the same material conditional

Abstract By the representations theorem it is known that a fuzzy relation R is a preorder if and only if R = inf μϵL I μ T , where L is the set of logical states of R. In this paper it is shown how, in almost all cases, both for the t -norm Min and archimedean t -norms T, it is possible to consider that the fuzzy sets μ are normalized. Furthermore a partial order between logical states, defined from the elemental preorders I μ T , is studied.