1. Linguistic Descriptions Adriano Joaquim de Oliveira Cruz NCE e IM/UFRJ adriano@nce.ufrj.br © 2003

2. @2003 Adriano Cruz NCE e IM - UFRJNo. 2 Summary n Introduction n Fuzzy variables n Fuzzy implication n Fuzzy composition and inference

3. @2003 Adriano Cruz NCE e IM - UFRJNo. 3 Introduction n In order to discuss about a phenomenon from the real world it is necessary to use a number fuzzy sets n For example, consider room temperature, one could use low, medium and high temperature n Note that these sets may overlap, allowing some temperatures belong partially to more than one set

4. @2003 Adriano Cruz NCE e IM - UFRJNo. 4 Introduction cont. n The process includes the definition of the membership functions n The Universe of discourse is also an important parameter

5. Fuzzy Variables

6. @2003 Adriano Cruz NCE e IM - UFRJNo. 6 Fuzzy Variable n A fuzzy variable is defined by the quadruple V = { x, l, u, m} n X is the variable symbolic name: temperature n L is the set of labels: low, medium and high universe of discourse n U is the universe of discourse n M are the semantic rules that define the meaning of each label in L (membership functions).

7. @2003 Adriano Cruz NCE e IM - UFRJNo. 7 Fuzzy Variable Example n X = Temperature n L = {low, medium, high} n U = {x  X | -70 o <= x <= +70 o } n M = 1.0 0.0 -70-60-50-40-30-20-10010203040506070 lowmediumhigh

8. @2003 Adriano Cruz NCE e IM - UFRJNo. 8 Membership Functions? n Subjective evaluation: The shape of the functions is defined by specialists n Ad-hoc: choose a simple function that is suitable to solve the problem n Distributions, probabilities: information extracted from measurements n Adaptation: testing n Automatic: algorithms used to define functions from data

9. @2003 Adriano Cruz NCE e IM - UFRJNo. 9 Variable Terminology n Completude: A variable is complete if for any x  X there is a fuzzy set such as  (x)>0 1.0 0.0 -70-60-50-40-30-20-10010203040506070 1.0 0.0 -70-60-50-40-30-20-10010203040506070 Complete Incomplete

10. @2003 Adriano Cruz NCE e IM - UFRJNo. 10 Partition of Unity n A fuzzy variable forms a partition of unity if for each input value x n where p is the number of sets to which x belongs n There is no rule to define the overlapping degree between two neighbouring sets n A rule of thumb is to use 25% to 50%

11. @2003 Adriano Cruz NCE e IM - UFRJNo. 11 Partition of Unity 1,0 0,0 -70-60-50-40-30-20-10010203040506070 0,5 1,0 0,0 -70-60-50-40-30-20-10010203040506070 0,5 Partition of Unity No Partition of Unity

12. @2003 Adriano Cruz NCE e IM - UFRJNo. 12 Partition of Unity cont n Any complete fuzzy variable may be transformed into a partition of unity using the equation

13. Implications

14. @2003 Adriano Cruz NCE e IM - UFRJNo. 14 Implications n If x  A then y  B. n P is a proposition described by the set A n Q is a proposition described by the set B n P  Q: If x  A then y  B

15. @2003 Adriano Cruz NCE e IM - UFRJNo. 15 Implications n Implication is the base of fuzzy rules n a  b = ¬a  b

16. @2003 Adriano Cruz NCE e IM - UFRJNo. 16 Implication as a Relation n The rule if x is A then y is B can be described as a relation n where  (x,y) is the relation we want to discover, for example ¬x  y n There are over 40 implication relations reported in the literature

17. @2003 Adriano Cruz NCE e IM - UFRJNo. 17 Interpretations of Implications n There are two ways of interpreting implication n p  q : meaning p is coupled to q and implication is a T-norm operator

18. @2003 Adriano Cruz NCE e IM - UFRJNo. 18 p is coupled with q n Commonly used T-norms are: n Mandami n Larson n Bounded Difference

19. @2003 Adriano Cruz NCE e IM - UFRJNo. 19 p entails q n These implication operators are generalisations of the material implications in two-valued logic as in n a  b = ¬a  b n a  b = ¬a  (a  b)

20. @2003 Adriano Cruz NCE e IM - UFRJNo. 20 P entails q cont n Goguen (1969) n Kurt Godel

21. @2003 Adriano Cruz NCE e IM - UFRJNo. 21 Fuzzy Inference n Fuzzy inference refers to computational procedures used for evaluating fuzzy rules of the form if x is A then y is B n There are two important inferencing procedures –Generalized modus ponens (GMP) - mode that affirms –Generalized modus tollens (GMT) – mode that denies

22. @2003 Adriano Cruz NCE e IM - UFRJNo. 22 Modus Ponens n If x is A then y is B n We know that x is A’ then we can infer that y is B’ n All men are mortal n All men are mortal (rule) n Socrates is a man (this is true) n So Socrates is mortal (as a consequence)

23. @2003 Adriano Cruz NCE e IM - UFRJNo. 23 Fuzzy Modus Ponens n If x is A then y is B n We know that x is A’ then we can infer that y is B’ n Tall men are heavy (rule) n John is tall (this is true) n So John is heavy (as a consequence)

24. @2003 Adriano Cruz NCE e IM - UFRJNo. 24 Modus Tollens n If x is A then y is B n We know that y is not B then we can infer that x is not A n All murderers owns axes (rule) n John does not own an axe (this is true) n So John is not a murderer (as a consequence)

25. @2003 Adriano Cruz NCE e IM - UFRJNo. 25 Fuzzy Modus Tollens n If x is A then y is B n We know that y is not B then we can infer that x is not A n All rainy days are cloudy (rule) n Today is not cloudy (this is true) n So Today is not raining (as a consequence)

26. @2003 Adriano Cruz NCE e IM - UFRJNo. 26 How to find the consequent n If x is A then y is B n This rule is an implication R(x,y) n If x is A’, we want to know whether y is B’ n This may be computed if we know R(x,y) B’= A’  R(x,y) n For example:  B’ (y)=  x [  A’ (x)   R (x,y)]