1. Fuzzy Logic and Approximate 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
Example with Lukaseiwicz implication

7. Conditional and Qualified Propositions

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
Knowledge Base
Inference Engine
Data Base (Fact)
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 Note: Step2: Calculate conclusion 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
Theorem
Condition of solution and Solution itself
If the condition does not satisfy, approximate solution should be considered.