# Publication: Computing with uncertainty truth degrees: a convolution-based degrees

2017

## Director

Acceso abierto / Sarbide irekia
Tesis doctoral / Doktoretza tesia

## Abstract

Fuzzy set theory can be seen as a body of mathematical tools exceptionally well-suited to deal with incomplete information, unsharpness and non-stochastic uncertainty. Indeed, as a tool for translating natural imprecise human language into mathematical objects, fuzzy sets are playing a crucial role in engineering for bridging the gap between man and computers. However, it is widely spread that the assignation of an exact membership degree is not an easy task. As a possible solution to this difficulty, several generalizations of fuzzy sets have been introduced and studied in the literature. Moreover, these generalizations have shown to be very useful in many applications leading to improved results when generalizations of fuzzy sets are considered. Generalizations differ from fuzzy sets in the mathematical object used to model the imprecision and/or uncertainty. Specifically, fuzzy sets take elements in the unit interval [0, 1] while the generalizations use more intricate mathematical objects such as intervals (interval-valued fuzzy sets), subsets of the unit interval (set-valued fuzzy sets or hesitant fuzzy sets), or functions (type-2 fuzzy sets), among others. Nevertheless, the use of the generalizations of fuzzy sets has a main drawback. Before applying any of the generalizations of fuzzy sets, it is necessary to adapt ad-hoc each theoretical notion to the corresponding mathematical object which represents the uncertainty in the considered application, i.e., it is necessary to redefine each theoretical notion from the unit interval [0, 1] to more intricate mathematical objects. Rather early in the history of fuzzy sets it became clear that the natural relationship between classical set theory and classical logic can be mimicked generating a natural relationship between fuzzy set theory and many-valued logic. This many-valued logic is nowadays called fuzzy logic. Similarly, each generalization of fuzzy sets constitutes a new fuzzy logical system. All these logical systems coincide in the sense that all of them model uncertainty, but they differ in the mathematical object which represents it. It can be easily seen that the same problem of fuzzy sets and its generalizations is found in the different fuzzy logics, i.e., although all the logical systems are akin, every theoretical notion has to be redefined for each logic. This problem, as well as the large number of these logical systems that model uncertainty, has led us to study whether or not it is possible to find a system that can encompass these logics and it has motivated us to propose a logical system that can model the uncertainty in a malleable way. Especially focusing on those logical systems that turn up from fuzzy theory, in this dissertation we propound a new logical system which retrieves multiple logical systems in the literature. The main advantages of the proposed logical system are that: it will avoid the excessive repetitions of theoretical notions; it will allow to adapt the applications to the most suitable type of fuzzy set or fuzzy logic in a simpler way. In this dissertation we present the semantics of the proposed logical system as well as a indepth study of the convolution operations, which are applied to define the disjunction and conjunction connectives of the logical system.

## Keywords

Conjuntos difusos, Generalizaciones, Incertidumbre, Lógica difusa, Fuzzy sets, Generalizations, Uncertainty, Fuzzy logic

## Department

Automática y Computación / Automatika eta Konputazioa

## Doctorate program

Programa de Doctorado en Ciencias y Tecnologías Industriales (RD 99/2011)
Industria Zientzietako eta Teknologietako Doktoretza Programa (ED 99/2011)

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