FUZZY IMPLICATION CAN BE ARBITRARILY COMPLICATED - A THEOREM

In fuzzy logic, there are several methods of representing implication in terms of Br, V, and inverted left perpendicular; in particular, exp licit representations define a class of S implications, implicit repre sentations define a class of R implications. Some reasonable implicati on operations have been proposed, such as Yager's a(b), that are diffi cult to represent as S or R implications. For such operations, a new c lass of representations has recently been proposed, called A implicati ons, for which the relationship between implications and the basic ope rations &, V, and inverted left perpendicular 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 ope rations, or they can, but we simply have not found these representatio ns? In this paper we show that yes, the complexity is necessary, becau se there are operations that cannot be represented in a simpler form. (C) 1998 John Wiley & Sons.