On a non-nested level-based representation of fuzziness

• Published in: Fuzzy sets and systems Vol. 192; pp. 159 - 175
• Main Authors: Sánchez, D., Delgado, M., Vila, M.A., Chamorro-Martínez, J.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-04-2012
• Subjects: Collection; Crisps; Fuzziness; Fuzzy; Fuzzy logic; Fuzzy set theories; Fuzzy set theory; Gradual numbers; Mathematical models; Representations; Semantics; Gradual numbers; Fuzziness; Fuzzy set theories
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/j.fss.2011.07.002

In this paper we describe a non-nested level-based representation of fuzziness, closely related to some existing models and concepts in the literature. Our objective is to motivate the use of this non-nested model by describing its theoretical possibilities, and illustrating them with some existing applications. From a theoretical point of view, we discuss the semantics of the representation, which goes beyond and has as a particular case fuzzy sets as represented by a collection of
α
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cuts
. In addition, the proposed operations on level-based representations, contrary to those of existing fuzzy set theories, satisfy all the properties of Boolean logic. We discuss the contributions of the representation and operation by levels to practical applications, in particular for extending crisp notions to the fuzzy case. In this respect, an important contribution of the proposal is that fuzzy mathematical objects (not only sets and the corresponding predicates) and operations are uniquely and easily defined as extensions of their crisp counterparts. In order to illustrate this claim, we recall level representations of quantities (gradual numbers) and their complementarity to fuzzy intervals (often inappropriately called fuzzy numbers).
► We provide a representation of fuzziness using a finite subset of levels in (0,1]. ► Fuzzy mathematical objects are an assignment of their crisp counterparts to levels. ► Contrary to fuzzy sets, crisp representatives are not necessarily nested between levels. ► Operations are performed on the representatives of the same level independently. ► Fuzzification of crisp objects/operations is unique and keep all the properties of the crisp case.