1. Fuzzy Logic The restriction of classical propositional calculus to a two- valued logic has created many interesting paradoxes over the ages. For example, the barber of Seville is a classic paradox (also termed as Russell’s barber). In the small Spanish town of Seville, there is a rule that all and only those men who do not shave themselves are shaved by a barber. Who shaves the barber? Another example comes from ancient Greece. Does the liar from Crete lie when he claims, “All Cretians are liars”? If he is telling the truth, then the statement is false. If the statement is false, he is not telling the truth.

2. Fuzzy Logic Let S: the barber shaves himself S’: he does not S  S’ and S’  S T(S) = T(S’) = 1 – T(S) T(S) = 1/2 But for binary logic T(S) = 1 or 0 Fuzzy propositions are assigned for fuzzy sets:

3. Fuzzy Logic Negation Disjunction Conjunction Implication [Zadeh, 1973]

4. Fuzzy Logic Example: = medium uniqueness = = medium market size = Then…

5. Fuzzy Logic

6. When the logical conditional implication is of the compound form, IF x is, THEN y is, ELSE y is Then fuzzy relation is: whose membership function can be expressed as:

7. Fuzzy Logic Rule-based format to represent fuzzy information. Rule 1: IF x is, THEN y is, where and represent fuzzy propositions (sets) Now suppose we introduce a new antecedent, say, and we consider the following rule Rule 2: IF x is, THEN y is

8. Fuzzy Logic

9. Suppose we use A in fuzzy composition, can we get The answer is: NO Example: For the problem in pg 127, let A’ = A B’ = A’  R = A  R = {0.4/1 + 0.4/2 + 1/3 + 0.8/4 + 0.4/5 + 0.4/6} ≠ B

10. Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs The extension of truth operations for tautologies, contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow identically those in classical logic. However, the use of partially true (or partially false) simple propositions in compound propositional statements results in new ideas termed quasi tautologies, quasi contradictions, and quasi equivalence. Moreover, the idea of a logical proof is altered because now a proof can be shown only to a “matter of degree”. Some examples of these will be useful.

11. Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs Truth table (approximate modus ponens) AB ABAB(A  (A  B))(A  (A  B))  B.3.2.7.3.7.3.8.3.8 Quasi tautology.7.2.3.7.8.7.8 Truth table (approximate modus ponens) AB ABAB(A  (A  B))(A  (A  B))  B.4.1.6.4.6.4.9.4.9 Quasi tautology.6.1.4.6.9.6.9

12. Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs The following form of the implication operator show different techniques for obtaining the membership function values of fuzzy relation defined on the Cartesian product space X × Y:

13. Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs The following common methods are among those proposed in the literature for the composition operation, where is the input, or antecedent defined on the universe X, is the output, or consequent defined on the universe Y, and is a fuzzy relation characterizing the relationship between specific inputs (x) and specific outputs (y): Refer fig on next slide…

14. Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs where f(.) is a logistic function (like a sigmoid or step function) that limits the value of the function within the interval [0,1] Commonly used in Artificial Neural Networks for mapping between parallel layers of a multi-layer network.

15. Fuzzy Rule-based systems Using fuzzy sets as a calculus to interpret natural language. It is vague, imprecise, ambiguous and fuzzy. Fundamental terms  atoms Collection of atomic terms  composite or set of terms An atomic term (a linguistic variable) can be interpreted using fuzzy sets. An atomic term  in the universe of natural language, X. Define a fuzzy set in the universe of interpretations or meanings, Y as a specific meaning of .

16. Fuzzy Rule-based systems  XY Mapping of a linguistic atom  to a cognitive interpretation

17. Fuzzy Rule-based systems Composite

18. Linguistic Hedges It increases contrast.

19. Precedence of the Operations 1Hedge, not 2and 3or Example: Suppose we have a universe of integers, Y = {1,2,3,4,5}. We define the following linguistic terms as a mapping onto Y: “small” = “large” =

20. Example (contd) Then we construct a phrase, or a composite term:  = “not very small and not very, very large” which involves the following set-theoretic operations: Suppose we want to construct a linguistic variable “intensely small” (extremely small); we will make use of the equation defined before to modify “small” as follows:

21. Example (contd) “Intensely small” =

22. Rule-based Systems IF-THEN rule based form Canonical Rule Forms 1.Assignment statements x = large, x  y 2. Conditional statements If A then B, If A then B, else C 3. Unconditional statements stop go to 5 unconditional can be If any conditions, then stop If condition C i, then restrict R i

23. Decomposition of Compound Rule Any compound rule structure can be decomposed and reduced to a number of simple canonical rules. The most commonly used techniques Multiple Conjunctive Antecedents If x is and, then y is Define The rule can be rewritten. IF THEN

24. Multiple Disjunctive Antecedents If x is or or … or then y is IF THEN

25. Condition Statements 1.IF THEN ( ELSE ) decomposed into: IF THEN orIF NOT THEN 2. IF (THEN ) unless decomposed into: IF THEN orIF NOT THEN NOT 3. IF THEN ( ELSE IF THEN ( )) decomposed into: IF THEN orIF NOT and THEN NOT 4. Nested IF-THEN rules IF THEN (IF, THEN ( )) becomes IF and THEN Each canonical form is an implication, and we can reduce the rules to a series of relations.

26. Condition Statements “likely” “very likely” “highly likely” “true” “fairly true” “very true” “false” “fairly false” “very false” Let  be a fuzzy truth value “very true” “true” “fairly true” “fairly false” “false” A truth qualification proposition can be expressed as: “x is is  ” or x is is  =

27. Aggregation of fuzzy rule The process of obtaining the overall consequent (conclusion) from the individual consequent contributed by each rule in the rule-base is known as aggregation of rules. Conjunctive System of Rules: Disjunctive System of Rules:

28. Graphical Technique of Inference If x 1 is and x 2 is then y is, k = 1,2,..., r Graphical methods that emulate the inference process and make manual computations involving a few simple rules. Case 1: inputs x 1, and x 2 are crisp. Memberships 1x 1 = input(i)  (x 1 ) =  (x 1 – input(i)) = 0otherwise 1x 2 = input(i)  (x 2 ) =  (x 2 – input(i)) = 0otherwise

29. Graphical Technique of Inference For r disjunctive rules: A 11 refers to the first fuzzy antecedent of the first rule. A 12 refers to the second fuzzy antecedent of the first rule.