Fuzzy Logic and Approximate Reasoning

1. Fuzzy Logic andApproximate Reasoning 1. Fuzzy Propositions 2. Inference from Conditional Propositions 3. Approximate Reasoning 4. Fuzzy Control

2. Fuzzy Proposition • Fuzzy Proposition: • The proposition whose truth value is [0,1] • Classification of Fuzzy Proposition • Unconditional or Conditional • Unqualified of Qualified • Focus on how a proposition can take truth value from fuzzy sets, or membership functions.

3. Fuzzy Proposition • Unconditional and Unqualified • Example:

4. Unconditional and Qualified Propositions • Truth qualified and Probability qualified • Truth qualified “Tina is young is very true” (See Fig. 8.2)

5. Unconditional and Qualified Propositions • Probability qualified (See Fig. 8.3) • Note: Truth quantifiers = “True, False” with hedges Probability quantifiers =“Likely, Unlikely” with hedges

6. Conditional and Unqualified Propositions • Conditional and Unqualified • Example with Lukaseiwicz implication

7. Conditional and Qualified Propositions • Conditional and Qualified

8. Fuzzy Quantifiers • Absolute Quantifiers • Fuzzy Numbers: about 10, much more than 100, at least 5

9. Fuzzy Quantifiers • Fuzzy Number with Connectives

10. Fuzzy Quantifiers • Relative Quantifier • Example: “almost all”, “about half”, ”most” • See Fig. 8.5

11. Linguistic Hedges • Modifiers • “very”, ”more or less”, “fairly”, “extremely” • Interpretation • Example: Age(John)=26 Young(26)=0.8 Very Young(26)=0.64 Fairly Young(26)=0.89

12. Inference from Conditional Fuzzy Propositions • Crisp Case (See Fig. 8.6 & Fig. 8.7)

13. Inference from Conditional Fuzzy Propositions • Fuzzy Case • Compositional Rule of Inference • Modus Ponen

14. Inference from Conditional Fuzzy Propositions • Modus Tollen • Hypothetical Syllogism

15. Approximate Reasoning • Expert System Expert User Knowledge Aq. Module Explanatory Interface Inference Engine Data Base (Fact) Knowledge Base Meta KB Expert System

16. Approximate Reasoning • Expert System • Knowledge Base (Long-Term Memory) • Fuzzy Production Rules (If-Then) • Data Base (Short-Term Memory) • Fact from user or Parameters • Inference Engine • Data Driven (Forward Chaining, Modus Ponen) • Goal Driven (Backward Chaining, Modus Tollen) • Meta-Knowledge Base • Explanatory Interface • Knowledge Acquisition Module

17. Fuzzy Implications • Crisp to fuzzy extension of implication • S-Implication from 1

18. Fuzzy Implications • R-Implications from 2 • QL-Implication from 3

19. Selection of Fuzzy Implication • Criteria • Modus Ponen • Modus Tollen • Syllogism • Some operators satisfies the criteria for 4 kinds of intersection (t-norm) operators

20. Multi-conditional AR • General Schema • Step1: Calculate degree of consistency

21. Multi-conditional AR • Step2: Calculate conclusion • Note: • Example: HIGH = 0.1/1.5m + 0.3/1.6m + 0.7/1.7m + 0.8/1.8m + 0.9/1.9m + 1.0/2m + 1.0/2.1m + 1.0/2.2m OPEN = 0.1/30° + 0.2/40° + 0.3/50° + 0.5/60° + 0.8/70° + 1.0/80° + 1.0/90° (if Completely OPEN is 90°)

22. Multi-conditional AR Fact: “Current water level is rather HIGH… around 1.7m, maybe.” rather HIGH = 0.5/1.6m + 1.0/1.7m + 0.8/1.8m + 0.2/1.9m If HIGH then OPEN : R(HIGH, OPEN) = A  B rather HIGH : A’ = rather HIGH -------------------------------- a little OPEN : B’ = a little OPEN

23. Multi-conditional AR

24. Multi-conditional AR • Interpretation of rule connection • Disjunctive • Conjunctive • 4 ways of inference

25. The Role of Fuzzy Relation Equations • Role • Theorem • Condition of solution and Solution itself • If the condition does not satisfy, approximate solution should be considered.