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The concept of a Fuzzy Logic is one that it is very easy for the illinformed to dismiss as trivial and/or insignificant. It refers not to a fuzziness of logic but instead to a logic of fuzziness, or more specifically to the logic of fuzzy sets. Those that examined Lotfi A. Zadeh's concept more closely found it to be useful for dealing with real world phenomena. From a strictly mathematical point of view the concept of a Fuzzy Set is a brilliant generalization of the classical notion of a Set. Now the concept of a Fuzzy Set is well established as an important and practical construct for modeling. Moreover, Zadeh's formulation makes one realize how artificial is the classical blackwhite formulation of Aristotelian logic (Is A or Is NotA). In a world of shades of gray a blackwhite dichotomy involves an unnecessary arbitrariness, an artificiality imposed upon that world.
The purpose of the material here is to present the mathematical structure of the concept of Fuzzy Sets. This generalization is achieved by way of the concept of the characteristic function for a set.
One way of defining a set A is in terms of its characteristic function μ_{A}(x). A point x belongs to set A if and only if μ_{A}(x)=1. A characteristic function is a function from some universal set U to the binary set {0,1}.
The set operations of union, intersection and complementation are defined in terms of characteristic functions as follows.
The other set theory constructs that are essential are:
As indicated above a characteristic function is a mapping from the universal set U to the set {0,1}. A fuzzy set is defined in terms of a membership function which is a mapping from the universal set U to the interval [0,1]. A characteristic function is a special case of a membership function and a regular set (a.k.a a crisp set) is a special case of a fuzzy set. Thus the concept of a fuzzy set is a natural generalization of the concept of standard set theory.
It remains to be proven whether the standard operations of standard set theory; i.e., union, intersection and complementation, have proper analogues in fuzzy set theory.
A membership function is a function from a universal set U to the interval [0,1]. A fuzzy set A is defined by its membership function φ_{A} over U.
The operation of union, intersection and complementation are defined exactly the same as they are for standard sets in terms of the characteristic function; i.e.;
Set inclusion and set equality have a natural definition for fuzzy sets; i.e.,
Of course any definitions can be posited; the question is whether the corresponding theorems that hold in standard set theory hold in fuzzy set theory. With the above definitions most standard set theory theorems carry over into fuzzy set theory.
Some of the more important elementary theorems of standard set theory are:
In the analysis below let φ_{A}, φ_{B} and φ_{C} be the membership functions for the fuzzy sets A, B, and C respectively. Furthermore, for any element of the universal set p,
The associativity and commutativity of fuzzy set union and intersection follow from the definition and the associativity and commutativity of the maximum and minimum functions; i.e.,
The distributivity properties also follow from properties of the maximum and minimum functions but the proof is a bit longer.
The righthand side of the first distributivity relation is (A ∩ B) ∪ (A ∩ C) which for fuzzy sets involves the evaluation of w = max(min(x,y),min(x,z)). If x is less either y or z then w = x. If x is between y<z then also w=x. If x is greater than either y or z then w = max(y,z). Thus w = min(x,max(y,z). This expression is equivalent to A ∩ (B ∪ C).
The righthand side of the second distributivity relation is (A ∪ B) ∩ (A ∪ C) which requires the evaluation of w = min(max(x,y),max(x,z)). As in the case of the previous distributivity relation the various cases can be evaluated. If x is greater than either y or z then w=x. If x is between y and z for y<z then w = min(x,z) = x. If x is less than either y or z then w = min(y,z). Thus w = max(x,min(y,z)) or in set terms A ∪ (B ∩ C).
The reflexity of complementation is easily established.
The null set for fuzz sets is the fuzzy set Φ for which the membership function is zero for all elements.
Below are examples of two fuzzy sets. They are constructed the basis of the distance of a point from a center. If the distance is less than a certain minimum the point is definitely in the set; i.e., the set membership function equals 1.0. If the distance is greater than a certain maximum the point is definitely not in the set; i.e., the set membership function equals 0.0. If the distance is between the minimum and maximum the set membership function is a linear function of the distance above the minimum, as shown below.
In the displays below the brightness of the color represents the value of the set membership function. Thus black represents the points not in the set.
The union of these two sets is shown below with the color of the points in the union set being violet.
The intersection of the two sets is:
Now that the intersection of two fuzzy sets has been displayed it is possible to present a visually more interesting display of the union of two sets. In this display the points which are in both sets are again displayed in violet.
The complement of the first set above (the one in red) is given below.
And the intersection of the first set and its complement is not empty, as is shown below.
Another difference between fuzzy set theory and regular set theory concerns the union of a set with its complement. In regular set theory the union of a set with its complement gives the universal set. This is not the case for a fuzzy set, as is shown below:
If the union of the first fuzzy set with its complement were the universal set then the rectangle above would be uniformly bright red.
Ironically, Lotfi Zadeh is a good example of fuzzy set membership. The question of Zadeh's ethnicity is difficult to answer sharply. His father was TurkishIranian (Azerbaijani) and his mother was Russian. His father was a journalist working in Baku, Azerbaijan in the Soviet Union. He served as a correspondent for Iranian newspapers while dealing exportimport trade. His mother was a pediatrician. Lotfi was born in Baku in 1921 and lived there until his family moved to Tehran in 1931.
Even Lotfi's name is now subject to a degree of uncertainty. The correct spelling is LOTFI, but there are numerous instances of the F and T being reversed to LOFTI, even in books about fuzzy logic. A google search for "lotfi zadeh" brings up 144,000 cases but a search for "lofti zadeh" brings up 17,000. However there were only 318 which had both spellings.
Lotfi's education commenced with his early years in Soviet Union. There he witnessed in the schools the messianic zeal of the true believers in communism. By 1931 anyone with any sense who could get out got out of the Soviet Union. When his parents moved the family to Tehran they put him in an American Presbyterian missionary school. He thus was subject to Russian, American and Iranian cultural influences with their accompanying religious zeals. He does not easily fit into any classification system. One thing however that can be said of Lotfi Zadeh; he has a brilliant, versatile mind.
He completed his degree in electrical engineering in 1942. Political conditions were in turmoil in the world in general and in Iran in particular. In 1943 Lotfi decided to emigrate to America. Against the odds he was able to achieve this goal inspite of wartime conditions. In America he decided to enroll in the Massachusetts Institute of Technology (MIT) for a master's degree in electrical engineering. This he completed in 1946. He immediate went into a doctoral program at Columbia University in New York, completing it in 1949. He became an assistant professor at Columbia upon his graduation. By 1957 he was a full professor at Columbia. In 1958 he received an offer from the University of California at Berkeley to join its department of electrical engineering. He accepted the offer and moved to Berkeley in 1958. By 1963 he was chairman of his department.
It was in 1964 that he formulated his generalization of the concept of a set. He worked out his ideas while visiting New York where his parents lived. There was not anything fuzzy about its formulation; it was precise, rigorous mathematics. The name fuzzy logic was an interesting and appropriate way to describe his concept but probably was detrimental to its acceptance. Some name such as continuum logic would have avoided the connotations of imprecision, but the name is part of the culture now and nothing can be done about it.
It is appropriate to point out here that illdefined boundaries between disjoint sets does not require any generalization of the classical notion of a set. Consider sets of points in a topological space. A point on the boundary between two sets is defined as a point such that any neighborhood of that point contains elements of both sets. Most examples of such spatial sets have crip boundaries. For example, for sets in a two dimensional space the boundary between a set and its complement set is a one dimensional curve. But this is just a special case. It is easy to define two dimensional sets such that the boundary is a two dimensional region rather than a one dimensional curve. A construction of this nature is depicted below. At each stage the middle third of the extensions are included in the other set. The process of course has to proceed ad infinitum. The first four stages are shown below along with what the ultimate stage would look like.
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For the fully constructed set any point in the middle vertical third of the box is a boundary point of the red and cyan sets. Thus the boundary between the two sets is not a line but instead a rectangle.
Having established that standard sets can have fuzzy boundary sets it is now possible to interpret set membership as average membership over a minimal window of a particular size. In other words, observation is always relative to a particular scale or minimum window size and what is observed is the average characteristics within that minimum window, which in remote sensing is called the pixel.
To illustrate this point consider one of the preliminary stages, say the fourth, in the above construction of a set with a fuzzy boundary. Consider square windows centered upon some point that is not in the red set. For relative large windows the set membership might be 75 percent red. As the window size decreases there is a point reached in which the window lies entirely within the zone that will be the boundary of the red set. For that window size and below for a range the set membership of the window is 50 percent red. When the window size shrinks to the point most of the window is in the cyan finger containing the selected point the set membership drops to 0 percent red.
In the above graph the macro threshold refers to the observation window size such that the window lies entirely within the "boundary zone". The micro threshold refers to the window size which lies entirely within the subset of the "boundary zone" containing the selected point. The higher the stage in the construction of the set with a fuzzy boundary the closer the micro threshold is to zero. Thus for the fully constructed fuzzy boundary set the limit of the set membership as the window size goes to zero is 50 percent no matter whether or not the point upon which the windows are centered is in or not in the red set. As indicated, although observation does depend upon the scale or window size there is an extended range over which observation is independent of window size.
Another way of stating the above interpretation is that observation never deals with the underlying set but instead deals with some partition of the set into subsets; i.e., pixels. For observation set membership is not the issue; instead the issue is allocation of subsets to sets.
The above material is one interpretation of the matter of fuzzy boundary sets which emphasizes the scaledependent nature of observation. This existence of an alternative to fuzzy set theory does not preclude its development of a rigorous generalization of classical standard set theory.
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