1. Artificial Intelligence CIS 342
The College of Saint Rose
David Goldschmidt, Ph.D.

2. Fuzzy Logic Expert knowledge often uses vague and inexact terms
Fuzzy Logic describes fuzziness by specifying degrees
e.g. degrees of height, speed, distance, temperature, beauty, intelligence, etc.

3. Fuzzy Logic Unlike Boolean logic, fuzzy logic is multi-valued
Fuzzy logic represents degrees of membership and degrees of truth
Things can be part true and part false at the same time

4. Linguistic Variables A linguistic variable is a fuzzy variable
e.g. the fact “John is tall” implies linguistic variable “John” takes the linguistic value “tall”
Use linguistic variables to form fuzzy rules:
IF ‘project duration’ is long
THEN risk is high
IF risk is very high
THEN ‘project funding’ is very low

5. Qualifiers & Hedges What about linguistic values with qualifiers?
e.g. very tall, extremely short, etc.
Hedges are qualifying terms that modify the shape of fuzzy sets
e.g. very, somewhat, quite, slightly, extremely, etc.

6. Representing Hedges

7. Representing Hedges

8. Representing Hedges

9. Representing Hedges

10. Linguistic Variables & Hedges
write a function or method called very() that modifies the degree of membership e.g. double x = very( tall( 185 ) );
Linguistic Variables & Hedges

11. Crisp Set Operations
Crisp set operations developed by Georg Cantor in the late 19th century:

12. Crisp Set Operations
Crisp set operations developed by Georg Cantor in the late 19th century (continued):

13. Fuzzy Set Operations Complement
To what degree do elements not belong to this set?
m¬A(x) = 1 – mA(x)

14. Fuzzy Set Operations Containment Which sets belong to other sets?
Each element of the fuzzy
subset has smaller membership
than in the containing set

15. mA∩B(x) = min[ mA(x), mB(x) ]
Fuzzy Set Operations
Intersection
To what degree is the element in both sets?
mA∩B(x) = min[ mA(x), mB(x) ]

16. mAB(x) = max[ mA(x), mB(x) ]
Fuzzy Set Operations
Union
To what degree is the element in either or both sets?
mAB(x) = max[ mA(x), mB(x) ]

17. Fuzzy Rules 1965 paper: “Fuzzy Sets” (Lotfi Zadeh)
Apply natural language terms to a formal system of mathematical logic
1973 paper outlined a new approach to capturing human knowledge and designing expert systems using fuzzy rules

18. Fuzzy Rules
A fuzzy rule is a conditional statement in the familiar form:
IF x is A
THEN y is B
x and y are linguistic variables
A and B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively

19. Linguistic Variables A linguistic variable is a fuzzy variable
e.g. the fact “John is tall” implies linguistic variable “John” takes the linguistic value “tall”
Use linguistic variables to form fuzzy rules:
IF ‘project duration’ is long
THEN ‘risk’ is high
IF risk is very high
THEN ‘project funding’ is very low

20. Fuzzy Expert Systems
A fuzzy expert system is an expert system that uses fuzzy rules, fuzzy logic, and fuzzy sets
Many rules in a fuzzy logic system will fire to some extent
If the antecedent is true to some degree of membership, then the consequent is true to the same degree

21. Fuzzy Expert Systems
Two distinct fuzzy sets describing tall and heavy:

22. Fuzzy Expert Systems
IF height is tall
THEN weight is heavy

23. Fuzzy Expert Systems Other examples (multiple antecedents):
e.g. IF ‘project duration’ is long
AND ‘project staffing’ is large
AND ‘project funding’ is inadequate
THEN risk is high
e.g. IF service is excellent
OR food is delicious
THEN tip is generous

24. Fuzzy Expert Systems Other examples (multiple consequents):
e.g. IF temperature is hot
THEN ‘hot water’ is reduced;
‘cold water’ is increased

25. Fuzzy Inference
Named after Ebrahim Mamdani, the Mamdani method for fuzzy inference is:
1. Fuzzify the input variables
2. Evaluate the rules
3. Aggregate the rule outputs
4. Defuzzify

26. Fuzzy Inference – Example
x, y, and z are linguistic variables
A1, A2, and A3 are linguistic values on X
B1 and B2 are linguistic values on Y
C1, C2, and C3 are linguistic values on Z
Fuzzy Inference – Example
Rule 1: IF x is A3 OR y is B1 THEN z is C1
Rule 2: IF x is A2 AND y is B2 THEN z is C2
Rule 3: IF x is A1 THEN z is C3
Rule 1: IF ‘project funding’ is adequate OR ‘project staffing’ is small THEN risk is low
Rule 2: IF ‘project funding’ is marginal AND ‘project staffing’ is large THEN risk is normal
Rule 3: IF ‘project funding’ is inadequate THEN risk is high

27. Fuzzy Inference – Example
1. Fuzzification
project funding
project staffing
inadequate
small
marginal
large

28. Fuzzy Inference – Example
2. Rule 1 evaluation
risk
project staffing
project funding
adequate
small
low

29. Fuzzy Inference – Example
2. Rule 2 evaluation
project funding
project staffing
risk
marginal
large
normal

30. Fuzzy Inference – Example
2. Rule 3 evaluation
risk
project funding
inadequate
high

31. Fuzzy Inference – Example
3. Aggregation of the rule outputs
risk
low
normal
high

32. Fuzzy Inference – Example
4. Defuzzification
e.g. use the centroid method in which a vertical line slices the aggregate set into two equal halves
How can we calculate this?

33. Fuzzy Inference – Example
4. Defuzzification
Calculate the centre of gravity (cog): x dx

34. Fuzzy Inference – Example
4. Defuzzification
Use a reasonable sampling of points

35. Applications of Fuzzy Logic
Why use fuzzy expert systems or fuzzy control systems?
Apply fuzziness (and therefore accuracy) to linguistically defined terms and rules
Lack of crisp or concrete mathematical models exist
When do you avoid fuzzy expert systems?
Traditional approaches produce acceptable results
Crisp or concrete mathematical models exist and are easily implemented

36. Applications of Fuzzy Logic
Real-world applications include:
Control of robots, engines, automobiles, elevators, etc.
Cruise-control in automobiles
Temperature control
Reduce vibrations in camcorders
Handwriting recognition, OCR
Predictive and diagnostic systems (e.g. cancer)