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290 11 Fuzzy Logic this chapter we will show that there is a strong link between set theory, logic, and geometry. A fuzzy set theory corresponds to fuzzy logic and the semantic of fuzzy operators can be understood using a geometric model. The geometric visualization of fuzzy logic will give us a hint as to the possible connection with neural networks. Fuzzy logic can be used as an interpretation model for the properties of neural networks, as well as for giving a more precise description of their performance. We will show that fuzzy operators can be conceived as generalized output functions of computing units. Fuzzy logic can also be used to specify networks directly without having to apply a learning algorithm. An expert in a certain ﬁeld can sometimes produce a simple set of control rules for a dynamical system with less eﬀort than the work involved in training a neural network. A classical example proposed by Zadeh to the neural network community is developing a system to park a car. It is straightforward to formulate a set of fuzzy rules for this task, but it is not immediately obvious how to build a network to do the same nor how to train it. Fuzzy logic is now being used in many products of industrial and consumer electronics for which a good control system is suﬃcient and where the question of optimal control does not necessarily arise. 11.1.2 The fuzzy set concept The diﬀerence between crisp (i.e., classical) and fuzzy sets is established by introducing a membership function. Consider a ﬁnite set X = {x1 , x2 , . . . , xn } which will be considered the universal set in what follows. The subset A of X consisting of the single element x1 can be described by the n-dimensional membership vector Z(A) = (1, 0, 0, . . . , 0), where the convention has been adopted that a 1 at the i-th position indicates that xi belongs to A. The set B composed of the elements x1 and xn is described by the vector Z(B) = (1, 0, 0, ..., 1). Any other crisp subset of X can be represented in the same way by an n-dimensional binary vector. But what happens if we lift the restriction to binary vectors? In that case we can deﬁne the fuzzy set C with the following vector description: Z(C) = (0.5, 0, 0, ..., 0) In classical set theory such a set cannot be deﬁned. An element belongs to a subset or it does not. In the theory of fuzzy sets we make a generalization and allow descriptions of this type. In our example the element x1 belongs to the set C only to some extent. The degree of membership is expressed by a real number in the interval [0, 1], in this case 0.5. This interpretation of the degree of membership is similar to the meaning we assign to statements such as “person x1 is an adult”. Obviously, it is not possible to deﬁne a deﬁnite age which represents the absolute threshold to enter into adulthood. The act of becoming mature can be interpreted as a continuous process in which the membership of a person to the set of adults goes slowly from 0 to 1. R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996

11.1 Fuzzy sets and fuzzy logic 291 There are many other examples of such diﬀuse statements. The concepts “old” and “young” or the adjectives “fast” and “slow” are imprecise but easy to interpret in a given context. In some applications, such as expert systems, for example, it is necessary to introduce formal methods capable of dealing with such expressions so that a computer using rigid Boolean logic can still process them. This is what the theory of fuzzy sets and fuzzy logic tries to accomplish. 1 young mature 0.8 old degree of membership 0.2 0 10 20 30 40 50 60 70 age Fig. 11.1. Membership functions for the concepts young, mature and old Figure 11.1 shows three examples of a membership function in the interval 0 to 70 years. The three functions deﬁne the degree of membership of any given age in the sets of young, adult, and old ages. If someone is 20 years old, for example, his degree of membership in the set of young persons is 1.0, in the set of adults 0.35, and in the set of old persons 0.0. If someone is 50 years old the degrees of membership are 0.0, 1.0, 0.3 in the respective sets. Deﬁnition 11. Let X be a classical universal set. A real function μA : X → [0, 1] is called the membership function of A and deﬁnes the fuzzy set A of X. This is the set of all pairs (x, μA (x)) with x ∈ X. A fuzzy set is completely determined by its membership function. Note that the above deﬁnition also covers the case in which X is not a ﬁnite set. The set of support of a fuzzy set A is the set of all elements x of X for which (x, μA (x)) ∈ A and μA (x) > 0 holds. A fuzzy set A with the ﬁnite set of support {a1 , a2 , . . . , am } can be described in the following way A = μ1 /a1 + μ2 /a2 + · · · + μm /am , where μi = μA (ai ) for i = 1, . . . , m. The symbols “/” and “+” are used only as syntactical constructors. R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996

292 11 Fuzzy Logic Crisp sets are a special case of fuzzy sets, since the range of the function is restricted to the values 0 and 1. Operations deﬁned over crisp sets, such as union or intersection, can be generalized to cover also fuzzy sets. Assume as an example that X = {x1 , x2 , x3 }. The classical subsets A = {x1 , x2 } and B = {x2 , x3 } can be represented as A = 1/x1 + 1/x2 + 0/x3 B = 0/x1 + 1/x2 + 1/x3 . The union of A and B is computed by taking for each element xi the maximum of its membership in both sets, that is: A ∪ B = 1/x1 + 1/x2 + 1/x3 The fuzzy union of two fuzzy sets can be computed in the same way. The union of the two fuzzy sets C = 0.5/x1 + 0.6/x2 + 0.3/x3 D = 0.7/x1 + 0.2/x2 + 0.8/x3 is given by C ∪ D = 0.7/x1 + 0.6/x2 + 0.8/x3 The fuzzy intersection of two sets A and B can be deﬁned in a similar way, but instead of taking the maximum we compute the minimum of the membership of each element xi to A and B. The maximum or minimum of the membership values are just one pair of possible deﬁnitions of the union and intersection operations for fuzzy sets. As we show later on, there are other alternative deﬁnitions. 11.1.3 Geometric representation of fuzzy sets Bart Kosko introduced a very useful graphical representation of fuzzy sets [259]. Figure 11.2 shows an example in which the universal set consists only of the two elements x1 and x2 . Each point in the interior of the unit square represents a subset of X. The convention is that the coordinates of the representation correspond to the membership values of the elements in the fuzzy set. The point (1, 1), for example, represents the universal set X, with membership function μA (x1 ) = 1 and μA (x2 ) = 1. The point (1, 0) represents the set {x1 } and the point (0, 1) the set {x2 }. The crisp subsets of X are located at the vertices of the unit square. The geometric visualization can be extended to an n-dimensional hypercube. Kosko calls the inner region of a unit hypercube in an n-dimensional space the fuzzy region. We ﬁnd here all combinations of membership values that a fuzzy set could assume. The point M in Figure 11.2 corresponds to the fuzzy set M = 0.5/x1 + 0.3/x2 . The center of the square represents the most diﬀuse of all possible fuzzy sets of X, that is the set Y = 0.5/x1 + 0.5/x2 . The degree of fuzziness of a fuzzy set can be measured by its entropy. In the geometric visualization, this corresponds inversely to the distance between R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996

11.1 Fuzzy sets and fuzzy logic {x } = (0,1) 293 X = (1,1) 2 Y 0,5 M 0,3 ∅ = (0,0) {x 1} = (1,0) 0,5 Fig. 11.2. Geometric visualization of fuzzy sets the representation of the set and the center of the unit square. The set Y in Figure 11.3 has the maximum possible entropy. The vertices represent the crisp sets and have the lowest entropy, that is, zero. Note that the fuzzy concept of entropy is mathematically diﬀerent from the entropy concept in physics or information theory. Some authors prefer to use terms like index of fuzziness [239] or also crispness, certitude, ambiguity, etc. [55]. With this caveat we adopt a preliminary deﬁnition of the entropy of a fuzzy set M as the quotient of the distance d1 (according to some metric) of the corner which is nearest to the representation of M to the distance d2 from the corner which is farthest away. Figure 11.3 shows the two relevant segments. The entropy E(M ) of M is therefore E(M ) = d1 . d2 According to this deﬁnition the entropy is bounded by 0 and 1. The maximum entropy is reached at the center of the square. The union or intersection of sets can be also visualized using this representation. The membership function for the the union of two sets A and B can be deﬁned as μA∪B (x) = max(μA (x), μB (x)) ∀x ∈ X (11.1) and corresponds to the maximum of the corresponding coordinates in the geometric visualization. The membership function for the intersection of two sets A and B is given by μA∩B (x) = min(μA (x), μB (x)) ∀x ∈ X. (11.2) Together with the points representing the sets A and B, Figure 11.4 shows the points which represent their union and intersection. R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996

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