1. Soft Computing
Fuzzy Logic

2. FUZZY LOGIC
Motivation
• Modeling of imprecise concepts
E.g. age, weight, height,…
• Modeling of imprecise dependencies (e.g. rules), e.g. if
Temperature is high and Oil is cheap then I will turn-on the
generator
• Origin of information
- Modeling of expert knowledge
-Representation of information extracted from
inherentedly imprecise data

3. FUZZY LOGIC
To quantify and reason about fuzzy or vague terms of natural
language
Example: hot, cold temperature
small, medium, tall height
creeping, slow, fast speed
Fuzzy Variable
A concept that usually has vague (or fuzzy) values
Example: age, temperature, height, speed

4. FUZZY LOGIC
Universe of Discourse
Range of possible values of a fuzzy variable
Example: Speed: 0 to 100 mph

5. FUZZY LOGIC
Fuzzy Set (Value)
Let X be a universe of discourse of a fuzzy variable and x be
its elements
One or more fuzzy sets (or values) Ai can be defined over X
Example: Fuzzy variable: Age
Universe of discourse: 0 – 120 years
Fuzzy values: Child, Young, Old
A fuzzy set A is characterized by a membership function
µA(x) that associates each element x with a degree of
membership value in A
The value of membership is between 0 and 1 and it
represents the degree to which an element x belongs
to the fuzzy set A

6. FUZZY LOGIC
Fuzzy Set (Value)
In traditional set theory, an object is either in a set or not in a
set (0 or 1), and there are no partial memberships
Such sets are called “crisp sets”

7. FUZZY LOGIC
Fuzzy Set Representation
Fuzzy Set A = (a1, a2, … an)
ai = µA(xi)
xi = an element of X
X = universe of discourse
For clearer representation
A = (a1/x1, a2/x2, …, an/xn)
Example: Tall = (0/5’, 0.25/5.5’, 0.9/5.75’, 1/6’, 1/7’, …)

8. FUZZY LOGIC
Fuzzy Set Representation
For a continuous set of elements, we need some function to
map the elements to their membership values
Typical functions: sigmoid, gaussian

9. FUZZY LOGIC
Formation of Fuzzy Sets
• Opinion of a single person
• Average of opinion of a set of persons
• Other methods (e.g. function approximation from data
by neural networks)
• Modification of existing fuzzy sets
- Hedges
- Application of Fuzzy set operators

10. FUZZY LOGIC
Formation of Fuzzy Sets
Hedges: Modification of existing fuzzy sets to account for
some added adverbs
Types:
Concentration (very)
Square of memberships Conc(µA(x)) = [µA(x)]2
reduces small memberships values
0.1 changes to 0.01 (10 times reduction)
0.9 changes to 0.81 (0.1 times reduction)
Example: very tall

11. FUZZY LOGIC
Formation of Fuzzy Sets
Dilation (somewhat)
Square root of memberships
Dil(µA(x)) = [µA(x)]1/2
increases small memberships values
0.09 changes to 0.3
0.81 changes to 0.9
Example: somewhat tall

12. FUZZY LOGIC
Fuzzy Sets Operations
Intersection (A  B)
In classical set theory the intersection of two sets contains
those elements that are common to both
In fuzzy set theory, the value of those elements in the
intersection:
µA  B(x) = min [µA(x), µB(x)]
e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)
Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75, 0/6)
Tall  Short = (0/5, 0.1/5.25, 0.5/5.5, 0.1/5.75, 0/6)
= Medium

13. FUZZY LOGIC
Fuzzy Sets Operations
Union (A  B)
In classical set theory the union of two sets contains those
elements that are in any one of the two sets
In fuzzy set theory, the value of those elements in the union:
µA  B(x) = max [µA(x), µB(x)]
e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)
Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75)
Tall  Short = (1/5, 0.8/5.25, 0.5/5.5, 0.8/5.75, 1/6)
= not Medium

14. FUZZY LOGIC
Fuzzy Sets Operations
Complement (A)
In fuzzy set theory, the value of complement of A is:
µ  A(x) = 1 - µA(x)
e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)
 Tall = (1/5, 0.9/5.25, 0.5/5.5, 0.2/5.75, 0/6)

15. FUZZY RULES
Fuzzy Rules
Relates two or more fuzzy propositions
If X is A then Y is B
e.g. if height is tall then weight is heavy
X and Y are fuzzy variables
A and B are fuzzy sets

16. FUZZY LOGIC
Fuzzy Relations
Classical relation between two universes
U = {1, 2} and V = {a, b, c} is defined as:
a b c
R=UxV= 1 1 1 1
2 1 1 1
Example:
U = Weight (normal, over)
V = Height (short, med, tall)

17. FUZZY LOGIC
Fuzzy Relations
Fuzzy relation between two universes U and V is defined as:
µR (u, v) = µAxB (u, v) = min [µA (u), µB (v)]
i.e. we take the minimum of the memberships of the two
elements which are to be related

18. FUZZY LOGIC
Fuzzy Relations
Example:
Determine fuzzy relation between A1 and A2
A1 = 0.2/x1 + 0.9/x2
A2 = 0.3/y1 + 0.5/y2 + 1/y3
The fuzzy relation R is
R = A1 x A2 = 0.2 x 0.3 0.5 1
0.9

19. FUZZY LOGIC
Fuzzy Relations
Example:
R = A1 x A2 = 0.2 x 0.3 0.5 1
0.9
= min(0.2, 0.3) min(0.2, 0.5) min(0.2, 1)
min(0.9, 0.3) min(0.9, 0.5) min(0.9, 1)
= 0.2 0.2 0.2
0.3 0.5 0.9

20. FUZZY LOGIC
Fuzzy Relations
R = R(A1, A2) A2
a23
= 0.2 0.2 0.2 (1.0
0.3 0.5 0.9 )
0.2
a22 0.9
(0.5
)
0.2 0.5
a21
(0.3
) 0.2 0.3
a11 a12 A1
(0.2) (0.9)

21. FUZZY RULES
Fuzzy Associative Matrix
So for the fuzzy rule:
If X is A then Y is B
We can define a fuzzy matrix M(nxp) which relates A to B
M=Ax B
It maps fuzzy set A to fuzzy set B and is used in the fuzzy
inference process

22. FUZZY RULES
Fuzzy Associative Matrix
Concept behind M
a1  b1 a1  b2 …
a2  b1 …
.
.
.
If a1 is true then b1 is true; and so on

23. FUZZY RULES
Approximate Reasoning
Example: Let there be a fuzzy associative matrix M for the
rule: if A then B
e.g. If Temperature is normal then Speed is medium
Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200]
B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]

24. FUZZY RULES
Approximate Reasoning: Max-Min Inference
Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200]
B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]
then
M= (0, 0) (0, 0.6) . . .
(0.5, 0) . . .
.
.
.
= 0 0 0 0 0
0 0.5 0.5 0.5 0 by taking the minimum
0 0.6 1 0.6 0 of each pair
0 0.5 0.5 0.5 0
0 0 0 0 0