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.