1 / 35

PART 8 Approximate Reasoning

FUZZY SETS AND FUZZY LOGIC Theory and Applications. PART 8 Approximate Reasoning. 1. Fuzzy expert systems 2. Fuzzy implications 3. Selecting fuzzy implications 4. Multiconditional reasoning 5. Fuzzy relation equations 6. Interval-valued reasoning. Fuzzy expert systems. 2.

tana
Download Presentation

PART 8 Approximate Reasoning

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. FUZZY SETS AND FUZZY LOGIC Theory and Applications PART 8Approximate Reasoning 1. Fuzzy expert systems 2. Fuzzy implications 3. Selecting fuzzy implications 4. Multiconditional reasoning 5. Fuzzy relation equations 6. Interval-valued reasoning

  2. Fuzzy expert systems 2

  3. Fuzzy implications Extensions of classical implications: 3

  4. Fuzzy implications S-implications • Kleene-Dienes implication • Reichenbach implication • Lukasiewicz implication 4

  5. Fuzzy implications S-implications • Largest S-implication 5

  6. Fuzzy implications Theorem 8.1 6

  7. Fuzzy implications R-implications • Gödel implication • Goguen implication 7

  8. Fuzzy implications R-implications • Lukasiewicz implication • the limit of all R-implications 8

  9. Fuzzy implications Theorem 8.2 9

  10. Fuzzy implications QL-implications • Zadeh implication • When i is the algebraic productanduis the algebraic sum. 10

  11. Fuzzy implications QL-implications • When iis the bounded difference and uis the bounded sum, we obtain the Kleene-Dienes implication. • When i = iminandu=umax 11

  12. Fuzzy implications Combined ones 12

  13. Fuzzy implications Axioms of fuzzy implications 13

  14. Fuzzy implications Axioms of fuzzy implications 14

  15. Fuzzy implications Axioms of fuzzy implications 15

  16. Fuzzy implications Theorem 8.3 16

  17. Selecting fuzzy implications Generalized modus ponens any fuzzy implication suitable for approximate reasoning based on the generalized modus ponens should satisfy (8.13) for arbitrary fuzzy sets A and B. 17

  18. Selecting fuzzy implications Theorem 8.4 18

  19. Selecting fuzzy implications Theorem 8.5 19

  20. Selecting fuzzy implications Generalized modus tollens Generalized hypothetical syllogism 20

  21. Multiconditional reasoning • general schema of multiconditional approximate reasoning The method of interpolation is most common way to determine B‘. It consists of the following two steps: 21

  22. Multiconditional reasoning 22

  23. Multiconditional reasoning 23

  24. Multiconditional reasoning four possible ways of calculating the conclusion B': Theorem 8.6 24

  25. Fuzzy relation equations • Suppose now that both modus ponens and modus tollens are required. The problem of determining R becomes the problem of solving the following system of fuzzy relation equation:

  26. Fuzzy relation equations Theorem 8.7

  27. Fuzzy relation equations If then is also the greatest approximate solution to the system (8.30). 27

  28. Fuzzy relation equations Theorem 8.8

  29. Interval-valued reasoning Let A denote an interval-valued fuzzy set. LA,UAare fuzzy sets called the lower bound and the upper bound of A. A shorthand notation of A( x ) When LA = UA, A becomes an ordinary fuzzy set.

  30. Interval-valued reasoning given a conditional fuzzy proposition (if - then rule) where A, B are interval-valued fuzzy sets defined on the universal sets X and Y. given a fact how can we derive a conclusion in the form 30

  31. Interval-valued reasoning view this conditional proposition as an interval-valued fuzzy relation R = [LR,UR], where It is easy to prove that LR(x, y) ≦ UR(x, y) and, hence, R is well defined. 31

  32. Interval-valued reasoning Once relation R is determined, it facilitates the reasoning process. Given A’ = [LA’,UA’], we derive a conclusion B’ = [LB’,UB’]by the compositional rule of inference where iis a t-norm and 32

  33. Interval-valued reasoning Example let a proposition be given, where Assuming that the Lukasiewicz implication

  34. Interval-valued reasoning

  35. Exercise 8 • 8.2 • 8.4 • 8.8 • 8.9

More Related