Fuzzy Logic Rule Base

Consider questions rose above. Firstly, consider, for illustration, the following fuzzy

IF-THEN rule containing an OR operation:

IF a₁ is A₁ AND a₂ is A₂ OR a₃ is A₃ AND a₄ is A₄ THEN b is B." By convention, it is understood in logic as

"(IF a₁ is A₁ AND a₂ is A₂) OR (a₃ is A₃ AND a₄ is A₄) THEN b is B."

With this understanding and convention, this is clear that this statement is Equivalent to the combination of the given two fuzzy IF-THEN rules as:

"IF a₁ is A₁ AND a₂ is A₂ THEN b is B." "IF a₃ is A₃ AND a₄ is A₄ THEN b is B."

Thus, the fuzzy logic OR operation is not necessary to employ: this may shorten a statement of a fuzzy IF-THEN rule; however it increases the format complexity of the rules.

Secondly, consider the fuzzy logic NOT operation. For a negative statement like "IF a is not A," one can always interpret it by a positive one "IF a¯ is A" or "IF a is A" where A means "not A" in logic theory and also "complement of A" in set theory. Moreover, the statement "a¯ is A" or "a is A" can be evaluated by

 μ_A (a¯) = μ_A (a) = 1 - μ_A (a)

Illustration 1

Specified a fuzzy logic implication statement

"IF a₁ is A₁ AND a₂ is not A₂ OR a₃ is not A₃ THEN b is B,"

How one can rewrite this as a set of Equivalent general fuzzy IF-THEN rules in the unified form?

Solution

One may firstly drop the fuzzy logic OR operation by rewriting the specified statement as

"IF a₁ is A1 AND a₂ is not A₂ THEN b is B,"

"IF a₃ is not A₃ THEN b is B,"

One may then drop the fuzzy logic NOT operation by rewriting them like

"IF a₁ is A₁ AND a¯₂ is not A₂ THEN b is B,"

"IF a¯₃  is A₃ THEN b is B,"

Finally, these two general fuzzy IF-THEN rules can be evaluated as follows

μ_A1(a₁) ∧ μ_A2 (a¯₂ ) ⇒ μ_B(b)

μ_A3  (a¯₃) ⇒ μ_B(b)

Hence, one only needs two general fuzzy IF-THEN rules, (1) and (2), and three membership ship values  μ_A1(a₁), μ_A2(a₂), μ_A3(a₃) to infer the conclusion "b is B," namely, b ∈ B, along with the truth values μ_B(b) calculated from the three specified membership values.

Every other fuzzy logic operations can be simply expressed and defined only by the AND and OR operations. They can be evaluated through min and the max operations as given below:

μA(a1 ) ∧ μA (a2) : min {μ1(a1), μA(a2)};

Among

 μ_A(a¯) ∧ μA   (a₂) : max {μ₁ (a₁ ), μ_A

μ_A (a ) = μ_A  (a) = 1 - μ_A(a) (a₂)};

 μ_A (a ⇒ b) = μ_A  (a) ⇒ μ_A (b) = min {1, 1 + μ_A(a) - μ_A(b)}

μ_A (a ⇔ b) = μ_A  (a) ⇔ μ_A (b) = 1 -| μ _A(a) - μ _A(b) |

Consequently, each finite combinations of these fuzzy logic operations can be expressed only by the AND and OR operations, hence in any finite fuzzy logic inference statement

IF ... THEN ...,

While the condition part "IF ..." can be expressed only by the AND and OR operations.

Consequently, a finite fuzzy logic implication can usually be described by a set of general or common IF-THEN rules containing merely the fuzzy logic AND operation, in the given generic form as:

(a) "IF a₁₁ is A₁₁ AND . . . AND a_1n is A_1n THEN b₁ is B₁."

(b) "IF a₂₁ is A₂₁ AND . . . AND a_2n is A_2n THEN b₂ is B₂."

                                    ..........

 (c) "IF a_m1 is A_m1 AND . . . AND a_mn is A_mn THEN b₂ is B₂."

This family of general or common fuzzy IF_THEN rules is mostly called a fuzzy logic rule base.

This is remarked that the number of components in all rule needs not to be the similar. If n = 2 however a rule has one component in the situation part, say as,

"IF a₁₁ is A₁₁ THEN b₁ is B₁,"

One cam formally rewrites this as

""IF a₁₁ is A₁₁ AND a₁₂ is I₁₂ THEN b₁ is B₁." Here I₁₂ is a fuzzy subset along with μ₁₂ (a) = 1 for all a ∈ I₁₂

Now, one actually inserts an "always true" or redundant condition into the "IF ... AND ..." part to fill in the gap of the statement. In a common discussion of the subject the format of a fuzzy logic rule base can be remained simple.

This is clear that such general forms of a fuzzy logic rule base includes the non-fuzzy logic rule base involve the non fuzzy case and the unconditional case with only "b is B" as special cases. Furthermore, this common fuzzy logic rule base with merely the fuzzy logic AND operation in the situation part too covers too unusual fuzzy logic implication statements, as like the one shown in the later illustration.