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.