Towards fast algorithms for processing type-2 fuzzy data: Extending Mendel’s algorithms from interval-valued to a more general case

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 alpha- 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. In his recent research, J. Mendel has shown that for the practically important case of interval-valued fuzzy sets, processing such sets can also be reduced to interval computations. In this paper, we show that Mendel's idea can be naturally extended to arbitrary type-2 fuzzy numbers.