Unit 2. Fuzzy Sets and Fuzzy Logical Operations. Fuzzy Logic I 21

Transkript

1 Unit 2 Fuzzy Sets and Fuzzy Logical Operations Fuzzy Logic I 2

2 What Do We Know So Far? We know that fuzzy logic is a generalized kind of logic which uses the unit interval [, ] as the set of truth values. We know that a fuzzy set on a universe X is represented by a membership function which maps each element x X to a degree of membership from the unit interval [, ] directly generalizing characteristic functions. Fuzzy Logic I 22

3 What Do We Need? In order to be able to proceed IF-THEN rules involving vague linguistic expressions which are modeled by fuzzy sets, we need to have proper generalizations of logical operations and an inference scheme. Let us start with the first question: How can we extend the classical logical operations,, to the unit interval [, ]? Fuzzy Logic I 23

4 Repetition: Basic Properties of Logical Operations. p q = q p, p q = q p (commutativity) 2. p (q r) = (p q) r, p (q r) = (p q) r (associativity) 3. p (q r) = (p q) (p r), p (q r) = (p q) (p r) (distributivity) 4. p = p, p = p (neutral elements) 5. p =, p = (absorption) 6. p p = p, p p = p (idempotence) 7. ( p) = p (involution) 8. (p q) = p q, (p q) = p q (De Morgan laws) 9. p p =, p p = (excluded middle) Questions: Which of them do we need? Which of them can we pre- Fuzzy Logic I 24

5 Standard Requirements for Fuzzy Conjunctions. Commutativity 2. Associativity 3. Non-decreasingness 4. is neutral element An operation fulfilling these requirements is called triangular norm (t-norm). Fuzzy Logic I 25

6 Triangular Norms A mapping T : [, ] 2 [, ] is a triangular norm (t-norm) if it has the following properties (for all x, y, z [, ]): Commutativity: T (x, y) = T (y, x) Associativity: T (x, T (y, z)) = T (T (x, y), z) Non-decreasingness: x y T (x, z) T (y, z) Neutral element: T (x, ) = x Fuzzy Logic I 26

7 The Four Standard t-norms T M (x, y) = min(x, y) T P (x, y) = x y T L (x, y) = max(x + y, ) x if y = T D (x, y) = y if x = otherwise Fuzzy Logic I 27

8 The Four Standard t-norms (Cont d) T M.4.4 T P T L.4.4 T D Fuzzy Logic I 28

9 Some Observations For all x, y [, ], we have: T D (x, y) T L (x, y) T P (x, y) T M (x, y) It is easy to check that T M is the largest possible t-norm and that T D is the smallest possible t-norm T M is the only t-norm fulfilling idempotence (T (x, x) = x) All except T D are continuous T P is the only differentiable one T P is the only one that is strictly non-decreasing Fuzzy Logic I 29

10 Examples of Intersections T M.5.5 T P.5.5 T L.5.5 T D.5.5 Fuzzy Logic I 3

11 Some Historical Background Triangular norms have emerged from the field of probabilistic metric spaces (Menger, Schweizer & Sklar) Have been studied much in the fields of associative functions and semigroups The introduction of t-norms into fuzzy set theory can be traced back to the late 97ies (Antony/Sherwood, Alsina/Trillas/Valverde, Höhle/Klement/Weber) Since then, a rich theory has developed, also as a sub-field of fuzzy logic and fuzzy set theory Fuzzy Logic I 3

12 Nilpotent Minimum T nm (x, y) = min(x, y) if x + y > otherwise Fuzzy Logic I 32

13 The Frank Family (T F λ ) λ [, ] T F λ (x, y) = T M (x, y) if λ = T P (x, y) if λ = T L (x, y) log λ ( + (λ x )(λ y ) λ ) if λ = if λ ], [ ], [ Fuzzy Logic I 33

14 The Frank Family (examples) λ = 9 λ = λ = Fuzzy Logic I

15 The Hamacher Family (T H λ ) λ [, ] T H λ (x, y) = T D (x, y) if λ = if λ = x = y = xy if λ [, [ and (λ, x, y) (,, ) λ+( λ)(x+y xy) Fuzzy Logic I 35

16 The Hamacher Family (examples) λ = λ = 2 λ = Fuzzy Logic I

17 The Schweizer-Sklar Family (Tλ SS) λ [, ] Tλ SS (x, y) = T M (x, y) if λ = T P (x, y) if λ = T D (x, y) if λ = ( ( max (x λ + y λ ), )) λ if λ ], [ ], [ Fuzzy Logic I 37

18 The Schweizer-Sklar Family (examples) λ = λ =.5 λ = Fuzzy Logic I

19 The Yager Family (T Y λ ) λ [, ] T Y λ (x, y) = T D (x, y) if λ = T M (x, y) max ( ( ( x) λ + ( y) λ) ) λ, if λ = if λ ], [ Fuzzy Logic I 39

20 The Yager Family (examples) λ = λ = 2 λ = Fuzzy Logic I

21 How To Construct Triangular Norms? If we take a strictly decreasing continuous mapping f : [, ] [, ] which satisfies f() =, then T f (x, y) = f ( min(f(), f(x) + f(y)) ) is a continuous triangular norm. It can be shown (Schweizer & Sklar) that continuous Archimedean t-norms are uniquely characterized in this way, where a t-norm is called Archimedean if, for all x, y ], [, there exists an n N such that T (x,..., x) < y. Fuzzy Logic I 4

22 How To Construct Triangular Norms? (cont d). Continuous Archimedean t-norms the generator of which fulfills f() = are strictly non-decreasing on ], ] 2 ; such strictly nondecreasing continuous t-norms are called strict; 2. Continuous Archimedean t-norms the generator of which fulfills f() < have the property that for all x ], [ there exists an n N such that T (x,..., x) < ; }{{} n times Continuous t-norms having this property are called nilpotent. This means that continuous Archimedean t-norms are either strict or nilpotent, and that this choice is solely determined by the value of f(). Fuzzy Logic I 42

23 Ordinal Sums Let (T α ) α A be a family of t-norms and (]a α, e α [) α A be a family of non-empty, pairwise disjoint open subintervals of [, ]. Then the following function T : [, ] 2 [, ] is a t-norm: ( a α + (e α a α ) T x a α α T (x, y) = eα aα, y a ) α if (x, y) [a α, e α ] 2, eα aα min(x, y) otherwise. Note that all continuous t-norms are uniquely characterized as ordinal sums of continuous Archimedean t-norms. Fuzzy Logic I 43

24 Understanding Ordinal Sums Graphically Fuzzy Logic I

25 Summary Triangular norms is a reasonable class of operations for modeling fuzzy conjunctions Triangular norms are delighting objects of study much is known, but much is still unknown It is worth knowing about the mathematical background, however, in practical applications you will most often suffice with the three basic t-norms T M, T P, and T L Fuzzy Logic I 45

26 Fuzzy Logic I 46 Strange Birds

27 Fuzzy Logic I 47 Strange Birds (cont d)

28 Fuzzy Negations A mapping N : [, ] [, ] is called a negation if it is non-increasing and fulfills the boundary conditions N() = and N() =. A negation is called strong if it is continuous and strictly decreasing (therefore, bijective). A strong negation is called strict if it is involutive, i.e. N(N(x)) = x Fuzzy Logic I 48

29 Fuzzy Negations: Examples By far, the most common negation is the so-called standard negation N S (x) = x. It is easy to show that N S is a strict negation. Not very useful for practical applications, but of theoretical interest the so-called intuitionistic negation: N I (x) = if x = otherwise Fuzzy Logic I 49

30 Strict Fuzzy Negations: Representation A negation N is strict if and only if there exists a strictly increasing continuous mapping ϕ : [, ] [, ] with ϕ() = and ϕ() = (a so-called [, ] automorphism) such that the following representation holds: N(x) = ϕ ( ϕ(x)) Fuzzy Logic I 5

31 Repetition: Basic Properties of Logical Operations. p q = q p, p q = q p (commutativity) 2. p (q r) = (p q) r, p (q r) = (p q) r (associativity) 3. p (q r) = (p q) (p r), p (q r) = (p q) (p r) (distributivity) 4. p = p, p = p (neutral elements) 5. p =, p = (absorption) 6. p p = p, p p = p (idempotence) 7. ( p) = p (involution) 8. (p q) = p q, (p q) = p q (De Morgan laws) 9. p p =, p p = (excluded middle) Questions: Which of them do we need? Which of them can we pre- Fuzzy Logic I 5

32 Standard Requirements for Fuzzy Disjunctions. Commutativity 2. Associativity 3. Non-decreasingness 4. is neutral element An operation fulfilling these requirements is called triangular conorm (t-conorm). Fuzzy Logic I 52

33 Triangular Conorms A mapping S : [, ] 2 [, ] is a triangular conorm (tconorm) if it has the following properties (for all x, y, z [, ]): Commutativity: S(x, y) = S(y, x) Associativity: S(x, S(y, z)) = S(S(x, y), z) Non-decreasingness: x y S(x, z) S(y, z) Neutral element: S(x, ) = x Fuzzy Logic I 53

34 The Four Standard t-conorms S M (x, y) = max(x, y) S P (x, y) = x + y x y S L (x, y) = min(x + y, ) x if y = S D (x, y) = y if x = otherwise Fuzzy Logic I 54

35 The Four Standard t-conorms (Cont d) S M.4.4 S P S L.4.4 S D Fuzzy Logic I 55

36 Some Observations For all x, y [, ], we have: S M (x, y) S P (x, y) S L (x, y) S D (x, y) It is easy to check that S D is the largest possible t-conorm and that S M is the smallest possible t-conorm S M is the only t-conorm fulfilling idempotence (S(x, x) = x) All except S D are continuous S P is the only differentiable one S P is the only one that is strictly non-decreasing Fuzzy Logic I 56

37 Examples of Unions S M.5.5 S P.5.5 S L.5.5 S D.5.5 Fuzzy Logic I 57

38 The Frank Family (S F λ ) λ [, ] S F λ (x, y) = S M (x, y) if λ = S P (x, y) if λ = S L (x, y) ( log λ + (λ x )(λ y ) ) λ if λ = if λ ], [ ], [ Fuzzy Logic I 58

39 The Hamacher Family (S H λ ) λ [, ] S H λ (x, y) = S D (x, y) if λ = if λ = and x = y = x+y+(λ 2)xy if λ ], [ and (λ, x, y) (,, ) +(λ )xy Fuzzy Logic I 59

40 The Schweizer-Sklar Family (Sλ SS) λ [, ] S M (x, y) if λ = S SS λ (x, y) = S P (x, y) if λ = S D (x, y) if λ = ( ( max (( x) λ + ( y) λ )) ), λ if λ ], [ ], [ Fuzzy Logic I 6

41 The Yager Family (S Y λ ) λ [, ] S Y λ (x, y) = S D (x, y) if λ = S M (x, y) ( (x min λ + y λ) ) λ, if λ = if λ ], [ Fuzzy Logic I 6

42 From t-norms to t-conorms and Back Let T be a t-norm and N be a strong negation. Then S(x, y) = N ( T (N(x), N(y)) ) is a t-conorm. Observe: this is nothing else but the second De Morgan law: N(S(x, y)) = T (N(x), N(y)) Correspondingly, if S is a t-conorm, then T (x, y) = N ( S(N(x), N(y)) ) is a t-norm, which corresponds to the first De Morgan law: N(T (x, y)) = S(N(x), N(y)) Fuzzy Logic I 62

43 De Morgan Triples and Dual Operations In case that N is a strict negation, the two De Morgan laws are equivalent. A triple (T, S, N) consisting of a t-norm T, a t-conorm S, and a strict negation N such that the two De Morgan laws are fulfilled, is called a De Morgan triple. We call a t-conorm S and a t-norm T dual to each other if they fulfill the De Morgan laws for the standard negation N S : S(x, y) = T ( x, y) T (x, y) = S( x, y) Fuzzy Logic I 63

44 Which Basic Properties of Logical Operations Hold Now?. p q = q p, p q = q p (commutativity) 2. p (q r) = (p q) r, p (q r) = (p q) r (associativity) 3. p (q r) = (p q) (p r), p (q r) = (p q) (p r) (distributivity) 4. p = p, p = p (neutral elements) 5. p =, p = (absorption) 6. p p = p, p p = p (idempotence) 7. ( p) = p (involution) 8. (p q) = p q, (p q) = p q (De Morgan laws) 9. p p =, p p = (excluded middle) Fuzzy Logic I 64

45 Operations on Fuzzy Sets Let X be a non-empty universe and (T, S, N) be a De Morgan triple. Then we can define the following three operations (with A, B fuzzy sets on X): T -intersection A T B: S-union A S B: N-complement N A: µ A T B(x) = T (µ A (x), µ B (x)) µ A S B(x) = S(µ A (x), µ B (x)) Fuzzy Logic I 65

46 Implications S-implication: For a t-conorm S and a negation N, we define I S,N (x, y) = S(N(x), y) Residual implication (R-implication): For a (left-)continuous t-norm T, we define T (x, y) = sup{u [, ] T (x, u) y} Fuzzy Logic I 66

47 Examples I SM,N S (x, y) = max( x, y) TM (x, y) = I SP,N S (x, y) = x + x y TP (x, y) = if x y y otherwise if x y otherwise y x I SL,N S (x, y) = min( x + y, ) TL (x, y) = min( x + y, ) Fuzzy Logic I 67

48 Examples (cont d) I TM,N S I TP,N S I TL,N S Fuzzy Logic I 68

49 Aggregation Operators A function A : n N[, ] n [, ] is called an aggregation operator if it has the following properties:. A(x,..., x n ) A(y,..., y n ) whenever x i y i for all i {,..., n} 2. A(x) = x for all x [, ] 3. A(,..., ) = and A(,..., ) = Fuzzy Logic I 69

50 Aggregation Operators: Examples All t-norms and t-conorms are aggregation operators (with the conventions T (x) = x and S(x) = x) All weighted arithmetic and geometric means are aggregation operators Another prominent example ordered weighted average (OWA) operators: Take a sequence (x,..., x n ) and sort it into a descending list ( x,..., x n ). Then the corresponding OWA operator for a weight vector w = (w,..., w n ) is defined as OWA w (x,..., x n ) = n wi x i. Fuzzy Logic I 7