1. CS621: Artificial Intelligence
Pushpak Bhattacharyya CSE Dept., IIT Bombay
Lecture 11– Fuzzy Logic: Inferencing

2. Representation of Fuzzy sets
Let U = {x1,x2,…..,xn}
|U| = n
The various sets composed of elements from U are presented as points on and inside the n-dimensional hypercube. The crisp sets are the corners of the hypercube.
μA(x1)=0.3
μA(x2)=0.4
(0,1)
(1,1) x2
(x1,x2)
U={x1,x2} x2
A(0.3,0.4)
(1,0)
(0,0) Φ x1 x1
A fuzzy set A is represented by a point in the n-dimensional space as the point {μA(x1), μA(x2),……μA(xn)}

3. Called the entropy of a fuzzy set
Degree of fuzziness
The centre of the hypercube is the “most fuzzy” set. Fuzziness decreases as one nears the corners
Measure of fuzziness
Called the entropy of a fuzzy set
Fuzzy set
Farthest corner
Entropy
Nearest corner

4. (0,1)
(1,1) x2
(0.5,0.5) A
d(A, nearest)
(0,0)
(1,0) x1
d(A, farthest)

5. Distance between two fuzzy sets
Definition
Distance between two fuzzy sets
L1 - norm
Let C = fuzzy set represented by the centre point
d(c,nearest) = | | + |0.5 – 0.0|
= 1
= d(C,farthest)
=> E(C) = 1

6. Definition
Cardinality of a fuzzy set
[generalization of cardinality of classical sets]
Union, Intersection, complementation, subset hood

7. How to define subset hood?
Note on definition by extension and intension
S1 = {xi|xi mod 2 = 0 } – Intension
S2 = {0,2,4,6,8,10,………..} – extension
How to define subset hood?

8. Meaning of fuzzy subset Suppose, following classical set theory we sayif
Consider the n-hyperspace representation of A and B
(0,1)
(1,1) A
Region where x2
. B1
.B2
.B3
(0,0)
(1,0) x1

9. This effectively means
CRISPLY
P(A) = Power set of A
Eg: Suppose
A = {0,1,0,1,0,1,…………….,0,1} – 104 elements
B = {0,0,0,1,0,1,……………….,0,1} – 104 elements
Isn’t with a degree? (only differs in the 2nd element)

10. Fuzzy definition of subset
Measured in terms of “fit violation”, i.e. violating the condition
Degree of subset hood = 1- degree of superset hood =
m(A) = cardinality of A

11. Show the relationship between entropy and subset hood Exercise 2:
We can show that
Exercise 1:
Show the relationship between entropy and subset hood
Exercise 2:
Prove that
Subset hood of B in A

12. Fuzzy sets to fuzzy logic
Forms the foundation of fuzzy rule based system or fuzzy expert system
Expert System
Rules are of the form If
then Ai
Where Cis are conditions
Eg: C1=Colour of the eye yellow
C2= has fever
C3=high bilurubin
A = hepatitis

13. In fuzzy logic we have fuzzy predicates
Classical logic
P(x1,x2,x3…..xn) = 0/1
Fuzzy Logic
P(x1,x2,x3…..xn) = [0,1]
Fuzzy OR
Fuzzy AND
Fuzzy NOT

14. Fuzzy Implication
Many theories have been advanced and many expressions exist
The most used is Lukasiewitz formula
t(P) = truth value of a proposition/predicate. In fuzzy logic t(P) = [0,1]
t( ) = min[1,1 -t(P)+t(Q)]
Lukasiewitz definition of implication

15. Eg: If pressure is high then Volume is low
High Pressure
Pressure

16. Fuzzy Inferencing

17. Fuzzy Inferencing: illustration through inverted pendulum control problem
Core
The Lukasiewitz rule
t( ) = min[1,1 + t(P) – t(Q)]
An example
Controlling an inverted pendulum θ
= angular velocity
i=current
Motor

18. The goal: To keep the pendulum in vertical position (θ=0)
in dynamic equilibrium. Whenever the pendulum departs from vertical, a torque is produced by sending a current ‘i’
Controlling factors for appropriate current
Angle θ, Angular velocity θ.
Some intuitive rules
If θ is +ve small and θ. is –ve small
then current is zero
If θ is +ve small and θ. is +ve small
then current is –ve medium

19. Control Matrix θ θ. -ve med -ve small +ve small +ve med Zero -ve med
Region of interest
Zero
Zero
+ve small
-ve small
Zero
+ve small
-ve small
-ve med
Zero
+ve med

20. Each cell is a rule of the form
If θ is <> and θ. is <>
then i is <>
4 “Centre rules”
if θ = = Zero and θ. = = Zero then i = Zero
if θ is +ve small and θ. = = Zero then i is –ve small
if θ is –ve small and θ.= = Zero then i is +ve small
if θ = = Zero and θ. is +ve small then i is –ve small
if θ = = Zero and θ. is –ve small then i is +ve small

21. Profiles Linguistic variables Zero +ve small -ve small 1 -ε3 ε3 -ε2 ε2+ε
Quantity (θ, θ., i)

22. Inference procedure Read actual numerical values of θ and θ.
Get the corresponding μ values μZero, μ(+ve small), μ(-ve small). This is called FUZZIFICATION
For different rules, get the fuzzy I-values from the R.H.S of the rules.
“Collate” by some method and get ONE current value. This is called DEFUZZIFICATION
Result is one numerical value of ‘i’.