Artificial Intelligence CIS 342

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

2. Fuzzy Logic • Expert knowledge oftenuses 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 membershipand 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 modifythe 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. write a function or method called very() thatmodifies the degree of membershipe.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. Fuzzy Set Operations • Intersection • To what degree is the element in both sets? mA∩B(x) = min[ mA(x), mB(x) ]

16. 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 • http://www.cs.berkeley.edu/~zadeh • 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 statementin 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 thatuses fuzzy rules, fuzzy logic, and fuzzy sets • Many rules in a fuzzy logic systemwill 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. 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 A3OR y is B1THEN z is C1 • Rule 2:IF x is A2AND y is B2THEN z is C2 • Rule 3:IF x is A1THEN z is C3 • Rule 1:IF ‘project funding’ is adequateOR ‘project staffing’ is smallTHEN risk is low • Rule 2:IF ‘project funding’ is marginalAND ‘project staffing’ is largeTHEN risk is normal • Rule 3:IF ‘project funding’ is inadequateTHEN 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 low adequate small

29. Fuzzy Inference – Example 2. Rule 2 evaluation risk project funding project staffing 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. x x dx x dx Fuzzy Inference – Example 4. Defuzzification • Calculate the centre of gravity (cog):

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) tolinguistically 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 andare 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 http://www.esru.strath.ac.uk/Reference/concepts/fuzzy/fuzzy_appl.de20.htm • Handwriting recognition, OCR • Predictive and diagnostic systems (e.g. cancer)