Date: 15-APR-93 Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth -- truth values between "completely true" and "completely false". It was introduced by Dr. Lotfi Zadeh of UC/Berkeley in the 1960's as a means to model the uncertainty of natural language. (Note: Lotfi, not Lofti, is the correct spelling of his name.) Zadeh says that rather than regarding fuzzy theory as a single theory, we should regard the process of ``fuzzification'' as a methodology to generalize ANY specific theory from a crisp (discrete) to a continuous (fuzzy) form (see "extension principle" in [2]). Thus recently researchers have also introduced "fuzzy calculus", "fuzzy differential equations", and so on (see [7]). Fuzzy Subsets: Just as there is a strong relationship between Boolean logic and the concept of a subset, there is a similar strong relationship between fuzzy logic and fuzzy subset theory. In classical set theory, a subset U of a set S can be defined as a mapping from the elements of S to the elements of the set {0, 1}, U: S --> {0, 1} This mapping may be represented as a set of ordered pairs, with exactly one ordered pair present for each element of S. The first element of the ordered pair is an element of the set S, and the second element is an element of the set {0, 1}. The value zero is used to represent non-membership, and the value one is used to represent membership. The truth or falsity of the statement x is in U is determined by finding the ordered pair whose first element is x. The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0. Similarly, a fuzzy subset F of a set S can be defined as a set of ordered pairs, each with the first element from S, and the second element from the interval [0,1], with exactly one ordered pair present for each element of S. This defines a mapping between elements of the set S and values in the interval [0,1]. The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate DEGREES OF MEMBERSHIP. The set S is referred to as the UNIVERSE OF DISCOURSE for the fuzzy subset F. Frequently, the mapping is described as a function, the MEMBERSHIP FUNCTION of F. The degree to which the statement x is in F is true is determined by finding the ordered pair whose first element is x. The DEGREE OF TRUTH of the statement is the second element of the ordered pair. In practice, the terms "membership function" and fuzzy subset get used interchangeably. That's a lot of mathematical baggage, so here's an example. Let's talk about people and "tallness". In this case the set S (the universe of discourse) is the set of people. Let's define a fuzzy subset TALL, which will answer the question "to what degree is person x tall?" Zadeh describes TALL as a LINGUISTIC VARIABLE, which represents our cognitive category of "tallness". To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset TALL. The easiest way to do this is with a membership function based on the person's height. tall(x) = { 0, if height(x) < 5 ft., (height(x)-5ft.)/2ft., if 5 ft. <= height (x) <= 7 ft., 1, if height(x) > 7 ft. } A graph of this looks like: 1.0 + +------------------- | / | / 0.5 + / | / | / 0.0 +-------------+-----+------------------- | | 5.0 7.0 height, ft. -> Given this definition, here are some example values: Person Height degree of tallness -------------------------------------- Billy 3' 2" 0.00 [I think] Yoke 5' 5" 0.21 Drew 5' 9" 0.38 Erik 5' 10" 0.42 Mark 6' 1" 0.54 Kareem 7' 2" 1.00 [depends on who you ask] Expressions like "A is X" can be interpreted as degrees of truth, e.g., "Drew is TALL" = 0.38. Note: Membership functions used in most applications almost never have as simple a shape as tall(x). At minimum, they tend to be triangles pointing up, and they can be much more complex than that. Also, the discussion characterizes membership functions as if they always are based on a single criterion, but this isn't always the case, although it is quite common. One could, for example, want to have the membership function for TALL depend on both a person's height and their age (he's tall for his age). This is perfectly legitimate, and occasionally used in practice. It's referred to as a two-dimensional membership function, or a "fuzzy relation". It's also possible to have even more criteria, or to have the membership function depend on elements from two completely different universes of discourse. Logic Operations: Now that we know what a statement like "X is LOW" means in fuzzy logic, how do we interpret a statement like X is LOW and Y is HIGH or (not Z is MEDIUM) The standard definitions in fuzzy logic are: truth (not x) = 1.0 - truth (x) truth (x and y) = minimum (truth(x), truth(y)) truth (x or y) = maximum (truth(x), truth(y)) Some researchers in fuzzy logic have explored the use of other interpretations of the AND and OR operations, but the definition for the NOT operation seems to be safe. Note that if you plug just the values zero and one into these definitions, you get the same truth tables as you would expect from conventional Boolean logic. This is known as the EXTENSION PRINCIPLE, which states that the classical results of Boolean logic are recovered from fuzzy logic operations when all fuzzy membership grades are restricted to the traditional set {0, 1}. This effectively establishes fuzzy subsets and logic as a true generalization of classical set theory and logic. In fact, by this reasoning all crisp (traditional) subsets ARE fuzzy subsets of this very special type; and there is no conflict between fuzzy and crisp methods. Some examples -- assume the same definition of TALL as above, and in addition, assume that we have a fuzzy subset OLD defined by the membership function: old (x) = { 0, if age(x) < 18 yr. (age(x)-18 yr.)/42 yr., if 18 yr. <= age(x) <= 60 yr. 1, if age(x) > 60 yr. } And for compactness, let a = X is TALL and X is OLD b = X is TALL or X is OLD c = not (X is TALL) Then we can compute the following values. height age X is TALL X is OLD a b c ------------------------------------------------------------------------ 3' 2" 65 0.00 1.00 0.00 1.00 1.00 5' 5" 30 0.21 0.29 0.21 0.29 0.79 5' 9" 27 0.38 0.21 0.21 0.38 0.62 5' 10" 32 0.42 0.33 0.33 0.42 0.58 6' 1" 31 0.54 0.31 0.31 0.54 0.46 7' 2" 45 1.00 0.64 0.64 1.00 0.00 3' 4" 4 0.00 0.00 0.00 0.00 1.00 For those of you who only grok the metric system, here's a dandy little conversion table: Feet+Inches = Meters -------------------- 3' 2" 0.9652 3' 4" 1.0160 5' 5" 1.6510 5' 9" 1.7526 5' 10" 1.7780 6' 1" 1.8542 7' 2" 2.1844 An excellent introductory article is: Bezdek, James C, "Fuzzy Models --- What Are They, and Why?", IEEE Transactions on Fuzzy Systems, 1:1, pp. 1-6, 1993. For more information on fuzzy logic operators, see: Bandler, W., and Kohout, L.J., "Fuzzy Power Sets and Fuzzy Implication Operators", Fuzzy Sets and Systems 4:13-30, 1980. Dubois, Didier, and Prade, H., "A Class of Fuzzy Measures Based on Triangle Inequalities", Int. J. Gen. Sys. 8. The original papers on fuzzy logic include: Zadeh, Lotfi, "Fuzzy Sets," Information and Control 8:338-353, 1965. Zadeh, Lotfi, "Outline of a New Approach to the Analysis of Complex Systems", IEEE Trans. on Sys., Man and Cyb. 3, 1973. Zadeh, Lotfi, "The Calculus of Fuzzy Restrictions", in Fuzzy Sets and Applications to Cognitive and Decision Making Processes, edited by L. A. Zadeh et. al., Academic Press, New York, 1975, pages 1-39. ================================================================Go Back Up