1. On fuzziness: thinking, language and information
2. Introduction to fuzzy set theory
3. Classic Set Theory
4. Fuzzy Set Theory
5. Fuzzy Systems
1. On fuzziness: thinking, language and information
“Fuzziness” is used as a visual metaphor of vagueness, inaccuracy, in opposition to sharpness or well-definiteness.
Unlike the traditional endeavour in science and philosophy for avoiding vagueness (consider, for instance, the centrality of “clearness” and “distinction” in Descartes and by extension in modern science and philosophy), there is a growing awareness of the fact that our knowledge of reality (or the information being conveyed or received about a concrete reality) contains a constitutive vagueness depending on the pragmatic situation in which this knowledge or information is immersed. The fundaments of this certainty can be found in the principles of statistical and quantum mechanics (holographic principle), mathematics (incompleteness) or scientific methodology (Pointcaré 1907).
In opposition to a negative assessment of vagueness, fuzzy logic has started to be considered as one of the fundamental features of cognitive systems, language and knowledge, allowing them to achieve a plasticity and dynamism, which are essential for the adaptation to changing environments. Thus, the robustness of the cognitive and linguistic system -far from being damaged by its fuzzy or blurred character- is rooted in such fuzziness (Kosko 1995, Pérez-Amat 2008).
Intending to give an account of the way of human reasoning, which is simply inaccurate, flexible, analogical…, fuzzy logic has been developed as a logical calculus, embracing the classical one -though distancing from its approaches, particularly in its way of rigid reasoning, which has been a fundamental character of mathematics since platonic times (Ferrater Mora 1994, 409s). This logical calculus has been successfully employed in both artificial intelligence and the so-called fuzzy control of industrial application, and it has been posed as a basis of a quantitative approach to semantic information (Pérez-Amat 2008).
But the fuzziness that can be ascribed to information does not only concern the semantic level, which depends upon the more or less contingent characteristics of human reasoning. On the contrary, the information that can be obtained from an observed reality is intrinsically fuzzy: the signals that can be received from some object are ultimately wave phenomena, which can only convey –due to its constitutive nature- a finite number of data over a bi-dimensional or superficial domain bounding the observed objects. That is, the whole wave phenomena outside the sources can be determined by a discrete distribution over a surface surrounding the object, therefore the dimension of the wave distribution in the surrounding space cannot be bigger than the dimension corresponding to such 2-dimensional discrete distribution. Thus, although the real extension of the observed object is 3-dimensional (volumetric) and it might be continuous, just a blurred projection of the observed object over a bounding surface can be achieved based upon the observation of the wave phenomena. In other words, the information that can be gathered about something being observed is constitutively fuzzy (Díaz Nafría 2008; Díaz Nafría & Pérez-Montoro, 2010a, 2010b).
2. Introduction to fuzzy set theory
The fuzzy set theory was initiated by Zadeh in the early 1960s (1964, 1965) (see Bellman et al. (1964)). In 1951, Menger (1951) explicitly used the fuzzy relation "max-product" but with probabilistic interpretation.
Since 1965, fuzzy set theory has been developed considerably by Zadeh and many other researchers. This theory was started to be implemented in a wide range of scientific environments.
There have been many books on fuzzy set theory as the mathematically one by Negoita and Ralescu (1975). There are also two research collections edited by Gupta et al. and Zadeh et al. (1975) and (1977).
Apart from the excellent research works of Zadeh, other introductory articles are those presented by Gusev and Smirnova (1973), Ponsard (1975), Kandel and Byatt (1978), Chang (1972), Gale (1975), Watanabe (1969), and Aizerman (1977).
There are several literature citations on fuzzy sets written by De Kerf (1975), Kandel and Davis (1978), Gaines and Kohout (1977) and Kaufmann (1980).
Mathematical formulas of the fuzzy sets theory will be presented in the following sections (§ 3, § 4, § 5). The basic definitions of classical sets, and the definitions and types of fuzzy sets, are revised. A detailed explanation of the operations between fuzzy sets, rules and norms-t-s are also carried out. The properties and the composition of fuzzy relations are reviewed. The characteristics and approximate reasoning are analyzed.
3. Classic Set Theory
A classical set is a collection of objects of any kind. What is called set theory was proposed by Georg Cantor (1845-1918), a German mathematician. In set theory, the set and the element are primitives. They are not defined in terms of other concepts. Let A be a set, "x ∈ A" means that x is an element in the set A and "x ∉ A" means that x does not belong to the set A. The set A is completely specified by the elements it contains. For example, there is no difference between a set which consists of 2, 3, 5 and 7 elements and a set of all prime numbers under 11.
Let X be a universe of discourse in which the set A is a subset, i.e.
In the classical set theory, any element x which belongs to X, belongs or not to the subset A clearly and undoubtedly, without any other option apart from these two ones.
Membership or not of an arbitrary element x to a subset A is given in most cases by checking whether or not a predicate that characterizes the subset A and gives rise to a bipartition of the universe of discourse X.
3.1 Membership Functions
The concept of belonging or not of an element to a set A can be expressed numerically by membership function, also sometimes called characteristic function. This function assigns a binary bit (1 or 0) to each element x of the universe of discourse as x belongs or not to the set A
any set A ⊂ X can be defined by the pairs which form each element x of the universe and its membership function, as follows:
3.2 Operations between sets
The three operations are shown in the following table.
Table 1: Operations between classical sets
4. Fuzzy Set Theory
In fuzzy set theory, classical sets are called crisp sets in order to distinguish them from fuzzy sets. Let A be a classical set defined in the universe X, then for any element x in X, x ∈ A or x ∉ A. In fuzzy set theory this property is widespread, therefore, in a fuzzy set A, it is not necessary that x ∈ A or x ∉ A.
In recent years several definitions have been introduced to present the generalization of property membership (Dubio 1987), (Pawlak 1985), (Shafer 1976), but it seems that fuzzy set theory is the most intuitive among the other theories and existing theorems.
The generalization is as follows.
4.1 Fuzzy Sets
We can define the characteristic function uA: X → (0,1) for any classic set as shown in equation (2). In fuzzy set theory, the characteristic function is generalized so that the membership function assigns a value for each x ∈ X in the interval [0,1] instead of two-element set (0,1). The set is based on this extended membership is called fuzzy sets.
Definition 1. Universe of Discourse is defined as the set X of possible values that can take the variable x. It can be represented as:
Definition 2. The membership function uA (x) of a fuzzy set A is as follows:
Thus, any element x in X has degree of membership μA(x) ∈ [0,1]. A is completely determined by:
Example 1. Suppose someone wants to describe a class of fast land animals like ostrich, cheetah, horse, spider, man, tortoise and hare. Some of these animals definitely belongs to this class, while others like the tortoise or the spider do not belong. But there is another group of animals where it is difficult to determine whether they are fast or not. Using a fuzzy set, the fuzzy set for fast animals is:
(Cheetah, 1), (Ostrich, 0.9), (Hare, 0.8), (Gazelle, 0.7), (Cat, 0.4)
i.e., the hare belongs with grade of 0.8, the gazelle with grade of 0.7 and the cat with 0.4 grade to the class of fast animals.
If we assume that C is a classical finite set (x1, x2, ..., xn), then an alternative notation is
where + is an enumeration.
A part from it, Zadeh proposed a more convenient notation for fuzzy sets.
Example 2. The set of all the fast animals, in equation (10), is described by:
1/Chetah + 0.9/Ostrich + 0.8/Hare +0.7/Gazelle
that is, one may describe the fuzzy set in equation (9) is as follows:
where the symbol of division is only a separator of sets of each pair, and the sum is the union operation between all elements of the set. The + fulfills the a / x + b / x = max (a, b) / x, i.e., if the same item has two different degrees of membership 0.8 and 0.6, then the membership degree is 0.8. Any discrete universe can be written as follows:
but when X is uncountable or continuous, the above equation is described as:
Equations (12) and (14) can written with the classical notation as follows:
Example 3. Figure 1 shows some fuzzy sets defined in the universe of discourse Age. The fuzzy set "young" represents the membership degree with respect to the parameter youth where individuals of every age can have.
Figure 1: An example of fuzzy sets
It can be seen that the fuzzy sets overlap, so that an individual might have a degree of membership in two groups: "young" and "mature", indicating that it has qualities associated with both sets. The membership degree of x in A , as noted above, is represented by μA (x). The fuzzy set A is the union of the degrees of membership for all points of the universe of discourse X, which can also be expressed as:
Under the notation of fuzzy sets, μA(x)/x is an element of set A. The operation ∫x represents the union of fuzzy elements μA(x)/x. The universes of discourse with discrete elements use the symbols + and Σ to represent the union operation.
It is commonly convenient to define a fuzzy set with the help of some formula so that, for example, all "young" could be expressed as:
Definition 3. The function Γ: X → [0,1] is a function of two parameters defined as follows:
This function can be seen in fig. 2
Figure 2: An example of the function Γ
Definition 4. Let A and B be two fuzzy sets defined respectively on the universe X and Y, and is the fuzzy relation R defined on X × Y. The support of a fuzzy set A is the classical set containing all the elements of A with the membership degrees that are not zero. This is defined by S (A).
The support of a fuzzy set A is defined as follows:
Definition 5. A fuzzy set A is convex if and only if X is convex and
Definition 6: The height of a fuzzy set A on X, denoted by Alt (A) is defined as:
A fuzzy set A is called normal, if Alt (A) = 1, is subnormal if Alt (A) <1.
In fuzzy control theory, it is usual to deal only with convex fuzzy sets.
Definition 7. Given a number α ∈ [0,1] and a fuzzy set A, we define the α-cut of A as the classical set Aα which has the following membership function:
In conclusion, the α-cut consists of those elements whose membership degree exceeds or equals the threshold α.
4.2 Operations between Fuzzy Sets
Operations such as equality, and the inclusion of two fuzzy sets are derived from classical set theory. Two fuzzy sets are equal if each element of the universe has the same degree of membership in each one of them. The fuzzy set A is a subset of fuzzy set B if every element of the universe has a membership degree lower in A than in B.
Definition 8. Two fuzzy sets are equal (A = B) if and only if
Definition 9. A is a subset of B (A ⊆ B) if and only if
The fuzzy sets can be operated with each other in the same way as the classical sets, since the former is a generalization of the latter. The interpretation with fuzzy sets is not as simple as traditional sets because they are used the characteristics of membership functions. It is possible to define operations like union, intersection and complement using the same membership functions. Zadeh proposed the following (Zadeh 1965):
Definition 10. The intersection between two fuzzy sets is represented as follows:
Definition 11. The union between two fuzzy sets is represented as follows:
Definition 12. The complement of a fuzzy set is represented as follows:
Definition 13. The product of two fuzzy sets A and B is defined as
Definition 14. The sum of two fuzzy sets A and B is defined as
Definition 15. a function n: [0,1] → [0,1] is said that it is a negation function if and only if it verifies the following properties:
1 - n (0) = 1, n (1) = 0 (boundary condition)
2 - n (x) ≤ n (y) if x ≥ y (monotone)
It also says that n is strict if and only if
3 - n (x) is continuous
4 - n (x) <n (y) if x> y ∀ x, y ∈ [0,1]
and is involutive if and only s
5 - n (n (x)) = x ∀ x ∈ [0,1]
4.3 T-Norms and S-Norms
In fact, the above definitions are quite arbitrary and could have been defined in many other ways. This includes considering more other general definitions for the operations between fuzzy sets in which they only have the same properties, similar to those seen in the classical sets theory. At present it is considered correct to define the intersection operator by any application t-norm and the union operator by any application s-norm (Schweitzer and Sklar 1961, 1963, Weber 1983), which are non-decreasing functions, so increasing one of the sets, also imply an increase its intersection or union.
Definición 16. triangular Norm
A triangular norm or t-norm is a function t: [0,1] × [0,1] → [0,1] which verifies the following properties:
The t-norms are used to express the intersection of fuzzy sets:
It can be said that the min operator is a t-norm.
Definition 17. Triangular conorm:
A triangular conorm is also called t-conorm or s-norm, is an application s: [0,1] × [0,1] → [0,1] that satisfies the following requirements:
The s-norms are used to express the union of fuzzy sets:
It can be concluded that the max operator is a t-conorm.
And an Archimedean t-conorm is strict if and only if
4.4 Properties of Fuzzy Sets
The laws and properties that fufill the classical sets are not always followed in the case of fuzzy sets. The following sections examine what laws verify the fuzzy sets and what not:
Commutative property: always verified, because the t-norms s-norms are commutative by definition.
5. Fuzzy Systems
5.1 Fuzzy Relations
As seen before, all the operations of union, intersection and complement, operate in a single universe of discourse. However, the Cartesian product allows the product of universes of discourse.
5.1.1 Cartesian Product
Let X and Y be any two universes of discourse. A fuzzy relation R between X and Y is defined as a fuzzy set whose universe is the Cartesian product X × Y. That is:
If A1 and A2 ⊂ X ⊂ Y, and if the Cartesian product of A12 and A is defined as:
It can also be expressed as:
Definition 18. Let X and Y be continuous universes of discourse. Then the function
is a binary fuzzy relation on X × Y. If X × Y are discrete universes, then
The integral denotes the sets of all tuples μR (x, y) / (x, y) on X × Y. It is also possible to express Equation (37) with ∫X∫Y μR(x,y)/(x,y), ,i.e., with double integral.
Definition 19. Let R and S be binary relations defined on X × Y. The intersection of R and S is defined by:
T-norm can be used rather than the minimum.
Definition 20. The union of R and S is defined by:
S-norm can be used rather than the maximum.
Definition 21. A projection of a fuzzy relation μR: X1 × ... × Xn → [0,1] on the universe of discourse Xi, is defined as
5.2 Composition of Relations
Let R be a fuzzy relation in the product X × Y and S forms another relationship in Y × Z.
Definition 22. The sup-min composition of these two relations, denoted by R ° S, is defined as the fuzzy relation in X × Z whose membership function is:
Definition 23. The inf-max composition, denoted by R × S, is defined as:
Definition 24. The sup-product composition as fuzzy relations in X × Z whose membership function is defined as:
If we generalize the minimum and the product by a t-norm and the maximum by a s-norm, respectively, the compositions are obtained sup-t and inf-s:
5.3 Approximate Reasoning
Unlike classical logic, in fuzzy logic, reasoning is not precise, but it occurs in an approximate manner. This means that one can infer a consequent although the rule antecedent is not completely verified (Approximate Reasoning). The higher the degree of compliance of the antecedent of the rule, the more approximate to the original rule the consequent part will be. The approximate reasoning is generally summarized, by extension of classical reasoning in the forms of "generalized modus ponens" and "generalized modus tollens.
premise 1: Premis of the rule:
x IS A*
premise 2: rule:
IF x IS A THEN y IS B
Consequent: y is B*
where A, B, A * and B * are fuzzy sets defined on the universes of discourse X, Y with membership function μA(x), μB(y), μA*(x) and μB*(y) respectively. This is the generalized modus ponens, which is reduced to the classical modus ponens when A = A * and B = B *.
The function of involvement is represented by a fuzzy relation in X × Y: R = A → B
This function can be defined in several ways. For example,
1 - Mamdani Implication:
With respect to fuzzy control this Implication is the most important. Its definition is based on the intersection operation as described above,
that can be represented as a t-norm
(49)2- Zadeh Implication:
The most widespread implication. It firstly solves If A then B if not A then C and then take A → B as a special case where C coincides with its universe of discourse,
which can be written as
Finally, the conclusion B * is a fuzzy set B *= A * ° (A → B) can be evaluated by a generalization of Modus Ponens proposed by Zadeh:
(54)A more general case is that a system composed of r1 rules, each of which is of the form SI x ES Ai1 ENTONCES y ES Bi1
Finally we analyze the case of rules with two antecedents. Let A, B and C defined fuzzy sets in X, Y and Z respectively. The rules are represented as follows:
premise 1: Premisa of the rule:
x IS A* E and IS B*
premise 2: rule:
SI x IS Ai1 E y IS Bi2 THEN z IS Ci1 i2
Consequent: z is C*
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