Fuzzy implication can be arbitrarily complicated: A theorem
Francisco G. Fernandez, Владик Крейнович · International Journal of Intelligent Systems · 1998
In fuzzy logic, there are several methods of representing implication in terms of &, ∨, and ¬; in particular, explicit representations define a class of S implications, implicit representations define a class of R implications. Some reasonable implication operations have been proposed, such as Yager's ab, that are difficult to represent as S or R implications. For such operations, a new class of representations has recently been proposed, called A implications, for which the relationship between implications and the basic operations &, ∨, and ¬ is even more complicated. A natural question is: Is this complexity really necessary? In other words, is it true that A operations cannot be described as S or R operations, or they can, but we simply have not found these representations? In this paper we show that yes, the complexity is necessary, because there are operations that cannot be represented in a simpler form. © 1998 John Wiley & Sons, Inc.