From processing interval-valued fuzzy data to general type-2: Towards fast algorithms

It is known that processing of data under general type-1 fuzzy uncertainty can be reduced to the simplest case - of interval uncertainty: namely, Zadeh's extension principle is equivalent to level-by-level interval computations applied to α-cuts of the corresponding fuzzy numbers. However, type-1 fuzzy numbers may not be the most adequate way of describing uncertainty, because they require that an expert can describe his or her degree of confidence in a statement by an exact value. In practice, it is more reasonable to expect that the expert estimates his or her degree by using imprecise words from natural language - which can be naturally formalized as fuzzy sets. The resulting type-2 fuzzy numbers more adequately represent the expert's opinions, but their practical use is limited by the seeming computational complexity of their use. It turns out that for the practically important case of interval-valued fuzzy sets, processing such sets can also be reduced to interval computations - and that this idea can be naturally extended to arbitrary type-2 fuzzy numbers.

I. Knowledge Processing and Fuzzy Uncertainty

In many practical situations, we only have expert estimates for the inputs . Sometimes, experts provide guaranteed bounds on , and even the probabilities of different values within these bounds. However, such cases are rare. Usually, the experts' opinion about the uncertainty of their estimates are described by (imprecise, “fuzzy”) words from natural language. For example, an expert can say that the value of the -th quantity is approximately equal to 1.0, with an accuracy most probably about 0.1. Based on such “fuzzy” information, what can we say about ?