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1. INTRODUCTION TO FUZZY LOGIC – Part 2 Dr. LINI MATHEW ASSOCIATE PROFESSOR, ELECTRICAL ENGINEERING DEPARTMENT NITTTR, CHANDIGARH

2. Three variables of interest in power transistors are the amount of current that can be switched, the voltage that can be switched and the cost. The following membership functions for power transistors were developed from hypothetical components catalog: Average current Average voltage ( a) Find the Cartesian Product P = VxI . FUZZY RELATIONS

3. The Cartesian Product expresses the relationship between V i and I j , where V i and I j are individual elements in the fuzzy set V and I. A fuzzy set is defined for the cost C in rupees, of a transistor (b)Using Fuzzy Cartesian Product, find T = IxC . (c) Using max-min composition find E = PoT (d) Using max-product composition find E = PoT FUZZY RELATIONS

4. FUZZY RELATIONS P =

5. FUZZY RELATIONS T =

6. FUZZY RELATIONS E =

7. Logic is but a small part of the human capacity to reason. It is the science of reasoning!! Logic for humans is a way quantitatively to develop a reasoning process that can be replicated and manipulated with mathematical precepts. LOGIC

8. Logic is the study of truth in logical propositions In classical logic, this truth is binary – a proposition is either true or false. In fuzzy logic, all truths are partial or approximate. LOGIC

9. Speed = 0 Speed = 1 Classical Logic [0.0 - 0.25] [0.25 – 0.50] [0.50 – 0.75] [0.75 – 1.00] Fuzzy Logic LOGIC

10. A simple proposition P is a linguistic , or declarative, statement contained within a universe of elements, X , that can be identified as being a collection of elements in X, which are strictly true or strictly false. Example : ( i ) A is B Temperature (A) is high (B) ( ii) (A is B) is C Anju (A) is beautiful (B) is very true (C) PROPOSITIONS

11. A proposition P is a collection of elements, A set, where the truth values for all elements in the set are either all true or all false. The veracity (truth) of an element in the proposition P can be assigned a binary truth value, called T(P ) PROPOSITIONS

12. For binary (Boolean) classical logic, T(P)is assigned a value of 1 (truth) or 0 (false). If U is the universe of all propositions, T hen T is a mapping of the elements, u, in these propositions (sets) to the binary quantities (0, 1), or T : u ∈ U -→ (0, 1). PROPOSITIONS

13. Five logical connectives to combine simple propositions on the same universe of discourse These connectives can be used to form new propositions from simple propositions. LOGICAL CONNECTIVES

14. disjunction ( ∨ ) conjunction ( ∧ ) negation ( − ) implication ( → ) equivalence ( ↔ ) Disjunction P ∨ Q : x ∈ A or x ∈ B ; T(P ∨ Q) =max(T(P), T(Q)). LOGICAL CONNECTIVES

15. Conjunction P ∧ Q : x ∈ A and x ∈ B; T(P ∧ Q) = min(T(P), T(Q)). Negation If T(P) = 1, then T(P̅) = 0; If T(P) = 0, then T(P̅) = 1. LOGICAL CONNECTIVES

16. Implication (P → Q) : x ∉ A or x ∈ B; T(P → Q) = T(P̅ ∪ Q). Equivalence (P ↔ Q) : T(P ↔ Q) = 1, for T(P) = T(Q) = 0, f or T(P) ≠ T(Q) LOGICAL CONNECTIVES

17. Example: F our propositions: 1. if 1 + 1 = 2, then 4 > 0; 2. if 1 + 1 = 3, then 4 > 0; 3. if 1 + 1 = 3, then 4 < 0; 4. if 1 + 1 = 2, then 4 < 0. The first three propositions are all true; the fourth is false. IMPLICATION

18. In the first two, the conclusion 4 > 0 is true regardless of the truth of the hypothesis; In the third case both propositions are false, but this does not disprove the implication; In the fourth case, a true hypothesis cannot produce a false conclusion IMPLICATION

19. The compound proposition P → Q is true in all cases except where a true antecedent P appears with a false consequent, Q, A true hypothesis cannot imply a false conclusion . IMPLICATION

20. Implication is true for all propositions of P and Q except for those propositions that are in both the truth set of P and the false set of Q. IMPLICATION

21. In set-theoretic form Linguistically equivalent to the statement “P → Q is true” when either “not A” or “B” is true IMPLICATION

22. LOGICAL CONNECTIVES

23. P is a proposition described by set A , which is defined on universe X , Q is a proposition described by set B , which is defined on universe Y . The implication operation : IF P , THEN Q P → Q represented as IF A, THEN B IMPLICATION

24. In set-theoretic terms, defined by the relation R, as: R = (A × B) ∪ ( Ā × Y) ≡ IF A, THEN B IF x ∈ A, where x ∈ X and A ⊂ X THEN y ∈ B, where y ∈ Y and B ⊂ Y, IMPLICATION

25. Another rule form: IF A, THEN B, ELSE C. or IF A, THEN B and IF Ā, THEN C. In classical logic, this rule has the form (P → Q) ∧ (P̅ → S); P : x ∈ A , A ⊂ X, Q : y ∈ B , B ⊂ Y and S : y ∈ C, C ⊂ Y . IMPLICATION

26. The set-theoretic equivalent of this is R = (A × B) ∪ (Ā × C) ≡ IF A, THEN B, ELSE C IMPLICATION

27. A fuzzy logic proposition, P is a statement involving some concept without clearly defined boundaries. Linguistic statements that tend to express subjective ideas and that can be interpreted slightly differently by various individuals typically involve fuzzy propositions . FUZZY LOGIC PROPOSITION

28. Most natural language is fuzzy, in that it involves vague and imprecise terms. The truth value assigned to P can be any value on the interval [0, 1]. The assignment of the truth value to a proposition is actually a mapping from the interval [0, 1] to the universe U of truth values, T T : u ∈ U → (0, 1 ). FUZZY LOGIC

29. Fuzzy propositions are assigned to fuzzy sets. A proposition P is assigned to fuzzy set A; then, T he truth value of a proposition, T(P) = µ A (x) where 0 ≤ µ A ≤ 1 FUZZY LOGIC

30. Disjunction P ∨ Q : x is A or B; T(P ∨ Q) =max(T (P), T (Q)). Conjunction P ∧ Q : x is A and B; T(P ∧ Q) = min(T (P), T (Q )). FUZZY LOGIC - CONNECTIVES

31. Negation T(P̅) = 1 - T(P ) Implication (P → Q) : x is A then x is B; T(P → Q) = T(P ̅ ∪ Q) = max(T(P̅), T (Q)). If x is A then y is B ie . R = (A x B) ∪(Ā x Y ) µ R ( x,y ) = max[( µ A (x ) ∧ µ B (y)), (1 - µ A (x ))] FUZZY LOGIC - CONNECTIVES

32. If x is A then y is B , else y is C R = (A x B) ∪(Ā x C) µ R ( x,y ) = max[(µ A (x) ∧ µ B (y)), (( 1 - µ A (x )) ∧ µ C (y))] FUZZY LOGIC - CONNECTIVES

33. Evaluating a new invention to determine its commercial potential. Two metrics to be used regarding the innovation of the idea. ( i ) uniqueness of the invention, (ii) market size of the invention’s commercial market, FUZZY INFERENCE

34. Uniqueness of the invention denoted by a universe of novelty scales, X = {1, 2, 3, 4 } Market size of the invention’s commercial market, denoted on a universe of scaled market sizes, Y = {1, 2, 3, 4, 5, 6 } FUZZY INFERENCE

35. Scores of a new invention Fuzzy set A = medium uniqueness Fuzzy set B = medium market size Fuzzy set C = diffuse market size Determine the implications ( i ) IF A, THEN B (ii) IF A, THEN B, ELSE C FUZZY INFERENCE

36. IF A, THEN B IF uniqueness is medium THEN market size is medium R = (A x B) ∪(Ā x Y ) µ R ( x,y ) = max[(µ A (x) ∧ µ B (y)), (1 - µ A (x))] FUZZY INFERENCE

37. FUZZY INFERENCE

38. FUZZY INFERENCE

39. FUZZY INFERENCE

40. FUZZY INFERENCE R = (A x B) ∪(Ā x Y)

41. IF A, THEN B, ELSE C IF uniqueness is medium THEN market size is medium , ELSE market size is diffuse R = (A x B) ∪(Ā x C ) µ R ( x,y ) = max[(µ A (x) ∧ µ B (y)), ((1 - µ A (x)) ∧ µ C (y))] FUZZY INFERENCE

42. FUZZY INFERENCE

43. FUZZY INFERENCE

44. Approximate Reasoning C onventional antecedent–consequent form, IF x is A, THEN y is B Suppose a new antecedent, A’, is introduced then IF x is A’, THEN y is B’ FUZZY INFERENCE

45. The consequent B ’ can be derived by means of fuzzy composition. B ’ =A’ ◦ R a nd R = (A x B) ∪(Ā x Y) FUZZY INFERENCE

46. The same invention example with the relation R which describes the invention’s commercial potential, the new antecedent A’ describes the uniqueness score of ‘almost high uniqueness’. Using max-min composition B’ = A’oR FUZZY RELATIONS

47. FUZZY RELATIONS B’=   This derived consequent B ’ is fairly diffuse, where there is no strong (or weak) membership value for any of the market size scores. ( ie . No membership value near 0 or 1)

48. Classical logical compound propositions that are always true, irrespective of the truth values of the individual simple propositions are called tautologies. Tautologies are useful for reasoning, for proving theorems, and for making deductive inferences . If a compound proposition can be expressed in the form of a tautology, the truth value of that compound proposition is known to be true. TAUTOLOGIES

49. Inference schemes in expert systems often employ tautologies because tautologies are formulae that are true on logical grounds alone. Some common tautologies are as follows:  B  X A  X ;  X  X (A ∧ (A → B)) → B (modus ponens ) ( (modus tollens )   TAUTOLOGIES

50. Modus Ponens Deduction, is a very common inference scheme used in forward-chaining rule-based expert systems. It is an operation whose task is to find the truth value of a consequent in a production rule, given the truth value of the antecedent in the rule. Modus ponens deduction concludes that given two propositions, P and P → Q, if both of which are true, then the truth of the simple proposition Q is automatically inferred. TAUTOLOGIES

51. Modus tollens inference, is used in backward-chaining expert systems. In modus tollens, an implication between two propositions is combined with a second proposition and both are used to imply a third proposition. TAUTOLOGIES

52. Proof of Modus Ponens TAUTOLOGIES

53. Truth Table of Modus Ponens TAUTOLOGIES

54. Proof of Modus Tollens TAUTOLOGIES

55. Truth Table of Modus Tollens TAUTOLOGIES

56. Contradictions Compound propositions that are always false , regardless of the truth value of the individual simple propositions constituting the compound proposition, are called contradictions. Eg . If A is the set of all prime numbers on the real line universe, X, then the proposition “A i is a multiple of 4” is a contradiction. TAUTOLOGIES

57. Equivalence Propositions P and Q are equivalent, that is, P ↔ Q, is true only when both P and Q are true or when both P and Q are false . Eg . the propositions P “triangle is equilateral” and Q “triangle is equiangular” are equivalent because they are either both true or both false for some triangle. The statement P ↔ Q is a tautology, if P is identical to Q, that is, if and only if T(P ) = T(Q ). TAUTOLOGIES

58. N atural language consists of fundamental terms characterized as atoms in literature . A collection of these atoms will form the molecules, or phrases, of our natural language. Atomic terms - examples : slow , medium, young, beautiful LINGUISTIC HEDGES

59. A collection of atomic terms is called composite terms Examples: very slow horse, young tree, medium-weight female, fairly beautiful painting . Can be modified with adjectives (nouns) or adverbs (verbs) These modifiers are called linguistic hedges LINGUISTIC HEDGES

60. Examples of linguistic hedges : very, low, slight , more or less, fairly, slightly, almost, barely, mostly, roughly, approximately etc. T he singular meaning of an atomic term is modified, or hedged, from its original interpretation. Modify the membership function for a basic atomic term. LINGUISTIC HEDGES

61. A basic linguistic atom be α, and when subject to some hedges. LINGUISTIC HEDGES

62. Linguistic terms are defined as a mapping onto Y = {1 , 2, 3, 4, 5} “Very small” = “small” 2 = LINGUISTIC HEDGES

63. “ Not Very Small” = 1 – “Very Small” “ Not Very Small and Not Very, Very Large ” LINGUISTIC HEDGES

64. Consider two universes of discourse, X and Y, and a functional transform (mapping) of the form y = f(x ). In universe X, there is fuzzy set A. The image of fuzzy set A on X under the mapping f is fuzzy set B , ie . through the same mapping, B = f(A ). EXTENSION PRINCIPLE

65. The membership functions describing A and B will now be defined on the universe of a unit interval [0, 1], and for the fuzzy case it becomes The image of fuzzy set A can be determined through the use of the composition operation, or B=A◦R, or when using the fuzzy vector form, b= a◦R where R is an n×m fuzzy relation matrix . EXTENSION PRINCIPLE

66. If the input universe comprises the Cartesian product of many universes. Let fuzzy sets A 1 ,A 2 , . . . ,A n be defined on the universes X 1 ,X 2 , . . . , X n . The mapping for these particular input sets can now be defined as B = f(A 1 ,A 2 , . . . ,A n ), where the membership function of the image B is given by ZADEH’S EXTENSION PRINCIPLE

67. Further generalizing the situation where a fuzzy input set, say A, maps to a fuzzy output through a fuzzy mapping, or B=f(A). The extension principle can be used to find the fuzzy image, B, by the following expression: ZADEH’S EXTENSION PRINCIPLE

68. Suppose there is a mapping between elements, u, of one universe, U, onto elements, v, of another universe, V, through a function f . Let this mapping be described by f : u → v. Define A to be a fuzzy set on universe U; ZADEH’S EXTENSION PRINCIPLE

69. For a function f that performs a one-to-one mapping i.e., maps one element in universe U to one element in universe V. ZADEH’S EXTENSION PRINCIPLE

70. For cases where this functional mapping f maps products of elements from two universes, say U 1 and U 2 , to another universe V, and we define A as a fuzzy set on the Cartesian space U 1 × U 2 , then : ZADEH’S EXTENSION PRINCIPLE

71. The complexity of the extension principle increases when we consider more than one of the combinations of the input variables, U 1 and U 2 , mapped to the same variable in the output space, V, that is, the mapping is not one-to-one. In this case, we take the maximum membership grades of the combinations mapping to the same output variable , ZADEH’S EXTENSION PRINCIPLE

72. Let a fuzzy set A be defined on the universe U = {1, 2, 3}. To map elements of A to another universe, V, under the function v = f(u ) = 2u−1 . The elements of V are V = {1, 3, 5}. The fuzzy set A is given as The fuzzy membership function for v = f (u) = 2u − 1 would be EXTENSION PRINCIPLE - EXAMPLES

73. Integers 1–10 are the elements of two identical but different universes, U 1 = U 2 = {1, 2, 3, . . . , 10}. Two fuzzy numbers A and B are defined on universes U1 and U2, respectively: A = “approximately 2” = B = “approximately 6” = EXTENSION PRINCIPLE - EXAMPLES

74. The product of (“approximately 2”) × (“approximately 6”) maps to a fuzzy number “approximately 12,” which is a fuzzy set defined on a universe, V, of integers, V = 5, 6, . . . , 18, 21 EXTENSION PRINCIPLE - EXAMPLES

75. Two fuzzy sets A and B, each defined on its own universe as follows: The membership values for the algebraic product mapping f(A,B ) = A×B (arithmetic product) EXTENSION PRINCIPLE - EXAMPLES

76. To map ordered pairs from input universes X 1 = {a, b} and X 2 = {1, 2, 3} to an output universe, Y = {x, y, z}. The mapping is given by the crisp relation, R, This relation represents a mapping, and it does not contain membership values. EXTENSION PRINCIPLE - EXAMPLES

77. A fuzzy set A is defined on universe X 1 and a fuzzy set B on universe X 2 as To determine the membership function of the output, C = f(A,B), whose relational mapping, f , is described by R. EXTENSION PRINCIPLE - EXAMPLES

78. C = f(A,B), whose relational mapping, f , is described by R. This is done using the extension principle μ C (x) = max[min(0.2, 0.6), min(0.2, 1), min(0.4 , 0.6)] = 0.4 μ C (y) = max[min(0.8, 1)] = 0.8, μ C (z) = max[min(0.8, 0.6), min(0.4, 1)] = 0.6. Hence EXTENSION PRINCIPLE - EXAMPLES

79. THANK YOU