Modeling of n-ary Fuzzy Relations and Operations on Them Using Multidimensional Matrices
Abstract
Fuzzy modeling is one of the widespread approaches to solving problems with difficult-to-formalize descriptions when we are dealing with so-called linguistic uncertainty, that is, uncertainty caused not by the stochastic nature of the phenomena being studied, but related to the fundamental features of human thinking and the reality around us, which consist in the fact that many concepts and phenomena are not They allow descriptions in the form of accurate estimates, but they are vague and indistinct. Examples of such concepts are "convenient road", "young man", "experienced programmer". To describe such phenomena and concepts, fuzzy sets can be used, as is known, the degree of membership of the elements to which can take any value belonging to the segment [0;1]. As in the case of ordinary sets, we can introduce the concept of a fuzzy relation by assigning to each tuple of elements of certain ordinary sets the meaning of belonging to a fuzzy relation. The properties of binary fuzzy relations are well known. One way to describe such relationships over them is to represent them as matrices.
In this paper, we propose to study the properties of n-ary (n > 2) fuzzy relations based on the theory of multidimensional matrices. The properties of multidimensional matrices are considered in detail in the monograph by N.P. Sokolov. Their application to the modeling of data processing processes is considered in the works of V.I. Munerman and his students (see, for example). In our opinion, this approach is a natural generalization of the theory of binary fuzzy relations and can be used in modeling complex systems and in building decision support systems.
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