Fuzzy IF-THEN Rules

Firstly, recall the fuzzy logic operations and, or, not, implication, and Equivalence also:

a ∧ b, a ∧ b, a¯, a ⇒ b, a ⇔ b,

And their evaluations on a fuzzy sets, A, with the membership function μ_A(.):

μ_A  (a ∧ b) : = min {μ_A  (a), μ_A  (b)} = μ_A (a) ∧ μ_A  (b);

μ_A  (a ∨ b) : = max {μ_A  (a), μ_A  (b)} = μ_A (a) ∨ μ_A  (b);

μ_A  (a ) = μ_A  (a) = 1 - μ_A  (a)

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) |

Recall the fuzzy relations also among elements of two fuzzy sets A and B, on which a membership function μ_A×B(a, b) is defined, along with a ∈ A and b ∈ B. This is clear that one can certainly consider the above fuzzy logic computations as several special fuzzy relations, along with A = B and μ_A×A = μ_A.

The implication relation a ⇒ b such can be interpreted, in linguistic terms, like

"IF a is true THEN b is true."

For fuzzy logic performed upon a fuzzy set A, there is a membership function μ_A explaining the truth values of a ∈ A and b ∈ A. In this case, a more complete linguistic statement would be as

"(IF a ∈ A is true with a truth value μ_A (a) THEN b ∈ A is true with a truth value μ_A (b)) has the truth value μ_A (a ⇒ b) = min {1, 1 + μA (b) - μA (a)}."

In the above, both a and b belongs to the similar fuzzy subset A and share the similar membership function μ_A. One has a non trivial fuzzy relation, which can be fairly complicated, if they belong to various fuzzy sets A and B with various membership functions μ_A and μ_B. However, In most cases, the implication relation a ⇒ b, performed on fuzzy sets A and B, here a ∈ A and b ∈ B, is simply defined in linguistic terms like

"(IF a ∈ A is true with a truth value μ_A (a) THEN b ∈ A is true with a truth vale μ_B (b)) has the truth value μ_A×B (a ⇒ b) = min {1, 1 + μ_B (b) - μ_A (a)}."

As all such statements have a standard format and their meaning is inside clear context, this is usual to write them in the following simple form as:

"IF a is A THEN b is B."

A fuzzy logic implication statement of this form is usually called a fuzzy IF- THEN rule. To be more usual, let A₁, A₂, . . . , A_n and B be the fuzzy subsets along with membership functions μ_A1, μ_A2, . . . , μ_An, and μ_B, respectively.

Definition of fuzzy logic.1

Commonly fuzzy IF- THEN rule has the form

"IF a1 is A1 AND...AND an is An THEN b is B."

By utilizing the fuzzy logic AND operation, such rule is implemented by the specified evaluation formula as:

 μ_AI (a1 ) ∧ ... ∧ (an) ⇒ μB (b) ,

 Here

 μ_Ai = (a₁ ) ∧ μ_Aj(a_j) = min {μ_A (a_i ), μ_A (a_j)}

And, thus,

μ_l(a₁) ∧ ... ∧ μ_A (a_n) = min{μ_A (a₁ ), ... , μ_A (a_n)}

About this general or common fuzzy IF-THEN rule and its evaluation, a few issues have to be clarified as:

• There is no fuzzy logic OR operation in a general or common fuzzy IF-THEN rule.

What should one do if a fuzzy logic implication statement includes the OR operation?

• There is no fuzzy logic NOT operation in a general or common fuzzy IF-THEN rule.