Publication Date
11-1-2023
Abstract
One of the ways to elicit membership degrees is by polling. For example, we ask a group of people how many believe that 30 C is hot. If 8 out of ten say that it is hot, we assign the degree 8/10 to the statement "30 C is hot". In precise mathematical terms, polling can be described via so-called random sets. It is known that every fuzzy set can be obtained this way, i.e., that every fuzzy set can be represented by an appropriate random set. Moreover, it is known that for many fuzzy sets, there are several different random-set representations. From the computational viewpoint, it is desirable to use the random sets which are the simplest, i.e., which contains the smallest possible number of elements. So, the natural questions are: what is the simplest random-set representation of a given fuzzy set? and is such simplest representation unique or are there several different random-set representations with the same number of elements? In this paper, we answer both questions: we show that for almost all fuzzy sense (in some reasonable sense), there are several different simplest random-set representations, and that the known α-cut representation (where probabilities are assigned to α-cuts of the fuzzy set) is one of them.
Comments
Technical Report: UTEP-CS-23-64