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

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.

Fuzzy Proposition Unconditional and Unqualified Example:

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

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

Conditional and Unqualified Propositions Example with Lukaseiwicz implication

Conditional and Qualified Propositions

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

Fuzzy Quantifiers Fuzzy Number with Connectives

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

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

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

Inference from Conditional Fuzzy Propositions Fuzzy Case Compositional Rule of Inference Modus Ponen

Inference from Conditional Fuzzy Propositions Modus Tollen Hypothetical Syllogism

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

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

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

Fuzzy Implications R-Implications from 2 QL-Implication from 3

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

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

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°)

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

Multi-conditional AR

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

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