In 1965, Zadeh presented the theory of fuzzy logic to present obscurity in linguistics and for more implementing and expressing human inference knowledge and ability in a natural way. Fuzzy logic begins by the theory of a fuzzy set. A fuzzy set is considered as a set has no a crisp defined boundary or limit. It can contain elements with a membership with partial degree.
A Membership Function (MF) is a plot that maps each input point in the space to a membership degree value, which is between 0, 1. Input space is usually named as the universe of discourse.
Let X be the universe of discourse and x be an inclusive part of X. A classical set A is considered as a group of elements x X, such that x may belong to or not belong to the set
A.
By describing a characteristic membership function on each element x in X, a classical set A can be defined by a ordered pairs set (x, 0) or (x, 1), where 1 indicates membership and 0, non membership.
The fuzzy set is different from the normal set in expressing the degree that an element becomes in a set. Hence, the feature of a fuzzy set membership is permit to own a value between 0 and 1, referring to the membership degree for an element in a certain set. If X is a group of elements pointed to by x, then if a fuzzy set A in X is related to ordered pairs set:
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A = {(x, µA (x))|x X} (3.4)
µA(x) is the membership function of the linguistic variable x in A that moves X in the
membership space M, for values between 0, 1. A is crisp and µA is coherent to the activation function in a crisp set.
Triangulated-shape and trapezoidal-shape membership functions are the most simple membership functions based on straight lines. Some of the other shapes are Gaussian, generalized bell, sigmoidal, and polynomial-based curves. Figure 3.2 illustrates the shapes of two commonly used MFs. The most significant issue to know about fuzzy logical reasoning is to understand of standard Boolean logic.
Fig. 3.2. Membership functions; (a) Gaussian and (b) trapezoidal.
3.2.2.1 Fuzzy Logic Operators
There is some coincidence between two-valued and multi-valued logic operations for the AND, OR, and NOT logical operators.
We can explain the expression A AND B, that A and B are between the range (0, 1) by using the operator minimum (A, B). By the way, we can alter the OR operator with the maximum operator, so that A OR B be equal to a maximum (A, B). Finally, the operation NOT A be equal to the operation 1–A.
There are some terms related to fuzzy logic operators such as fuzzy intersection or con- junction for AND operation, fuzzy union or dis-junction for OR operation, and fuzzy complement for NOT operation. The intersection between two fuzzy sets A and B is generally described by a binary mapping T that merges two membership functions as follows:
43 The fuzzy intersection operator is usually pointed to T-norm (Triangular norm) operator. The fuzzy union operator is generally addressed by a binary mapping S.
µA B (x) = S (µA (x),µB (x)) (3.6)
This class of fuzzy union operators is often referred to as T-conorm (S-norm) operators.
3.3.2.2 If-Then Rules and Fuzzy Inference Systems
The fuzzy rule base is featured in the formula of if-then rules in which antecedent and conclusion having linguistic variables. The fuzzy rules format the rule bases for the fuzzy logic system. Fuzzy if-then rules are often applied to achieve the imprecise states of reasoning that have an important part in the human capability to take decisions in an condition of imprecision and uncertainty. A single fuzzy if-then rule supposes the form
If x is A then y is B
A and B: linguistic values are known by fuzzy sets in the ranges of universes of
discourse, X and Y, respectively.
The “if” part of the rule ‘x is A’ is called the antecedent or precondition, while the “then” part of the rule ‘y is B’ is called the consequent or conclusion. Explaining an if-then rule has estimating the antecedent fuzzification of the input and activating any necessary fuzzy operators and then applying that result to the consequent known as implication. For rules with multiple antecedents, the terms of the antecedent are calculated simultaneously and merged to a single value by the logical operators. Similarly, all the consequents rules with multiple consequents are based equally on the antecedent result. The consequent defines a fuzzy set be addressed to the output.
Then Implication function upgrades the fuzzy set to the designed degree by the antecedent. For multiple rules, each rule output is a fuzzy set. The fuzzy outputs, defined for each rule, are then merged into a single output fuzzy set. Then, the resulted set is defuzzified to a single term.
The defuzzification interface is a map for a space of fuzzy processes known with an output universe of discourse to a space of non-fuzzy processes, due to the output from the inference engine is often a fuzzy set, but in most real problems, crisp numbers are the desired. The most famous defuzzification methods are maximum criterion, center of gravity, and the mean of maximum. The maximum criterion is the simplest one to apply. It develops the state for the possible distribution for achieving a maximum value [65] – [71].
It is basically benefit if the fuzzy rule base is adjustable to a certain application. The fuzzy rule base is often installed in manual or automatic way by some learning methods using evolutionary algorithms and/or neural network learning techniques.