Fuzzy Sets and Fuzzy Logic

Fuzzy Sets and Fuzzy Logic

Crisp sets Collection of definite, well-definable objects (elements). Representation of sets: list of all elements ={x,,x n }, x j X elements with property P ={x x satisfies P},x X Venn diagram X characteristic function f : X {,}, f (x) =, x f (x) =, x Real numbers larger than 3: 3 X

Crisp (traditional) logic Crisp sets are used to define interpretations of first order logic If P is a unary predicate, and we have no functions, a possible interpretation is = {,,2} P I = {,2} within this interpretation, P() and P(2) are true, and P() is false. Crisp logic can be fragile : changing the interpretation a little can change the truth value of a formula dramatically.

Sets with fuzzy, gradual boundaries (Zadeh 965) fuzzy set in X is characterized by its membership function µ : X [,] fuzzy set is completely determined by the set of ordered pairs ={(x,µ (x)) x X} X is called the domain or universe of discourse Real numbers about 3: 3 µ (x) X

N u m b e r o f C h i l d r e n Fuzzy sets on discrete universes Fuzzy set C = desirable city to live in X = {SF, Boston, L} (discrete and non-ordered) C = {(SF,.9), (Boston,.8), (L,.6)} Fuzzy set = sensible number of children X = {,, 2, 3, 4, 5, 6} (discrete universe) = {(,.), (,.3), (2,.7), (3, ), (4,.6), (5,.2), (6,.)} M e m b e r s h i p G r a d e s

Fuzzy set B = about 5 years old X = Set of positive real numbers (continuous) B = {(x, µ B (x)) x in X} µ B ( x) = + x 5 2 M e m b e r s h i p G r a d e s g e

Membership Function formulation Triangular MF: trimf ( x ; a,b, c)= max ( min ( x a b a, c x c b ), ) Trapezoidal MF: Gaussian MF: trapmf ( x ;a,b,c,d )=max ( min ( x a b a gaussmf ( x ;a,b)=e 2 ( x a b ) 2,, d x d c ), ) Generalized bell MF: gbellmf ( x; a, b, c) = 2 x c + b a

MF formulation (a) Triangular MF (b) Trapezoidal MF M em bership Grades.8.6.4.2 M em bership Grades.8.6.4.2 2 4 6 8 2 4 6 8 (c) Gaussian MF (d) Generalized Bell MF M em bership Grades.8.6.4.2 M em bership Grades.8.6.4.2 2 4 6 8 2 4 6 8

Fuzzy sets & fuzzy logic Fuzzy sets can be used to define a level of truth of facts Fuzzy set C = desirable city to live in X = {SF, Boston, L} (discrete and non-ordered) C = {(SF,.9), (Boston,.8), (L,.6)} corresponds to a fuzzy interpretation in which C(SF) is true with degree.9 C(Boston) is true with degree.8 C(L) is true with degree.6 membership function can be seen as a (fuzzy) predicate.

Notation Many texts (especially older ones) do not use a consistent and clear notation X is discrete X is continuous = x x = µ ( ) / µ ( i ) / i x x xi X X = µ ( xi ) xi = µ ( x) x xi X X Note that Σ and integral signs stand for the union of membership grades; / stands for a marker and does not imply division.

Fuzzy partition Fuzzy partition formed by the linguistic values young, middle aged, and old : M e m b e r s h i p G r a d e s. 5 Y o u n g M i d d l e g e d O l d 2 3 4 5 6 7 8 9 g e

Support, core, singleton The support of a fuzzy set in X is the crisp subset of X whose elements have non-zero membership in : supp() = {x X µ (x)>} The core of a fuzzy set in X is the crisp subset of X whose elements have membership in : core() = {x X µ (x)=} µ B Singleton fuzzy set Core Support X

Normal fuzzy sets The height of a fuzzy set is the maximum value of µ (x) fuzzy set is called normal if its height is, otherwise it is called sub-normal µ height() B is normal, B is sub-normal X

Set theoretic operations /Fuzzy logic connectives Subset: Complement: B µ µ B (Specific case) = X µ ( x) = µ x ( ) Union: C = B Intersection: C = B µ ( x) = max( µ ( x), µ ( x)) = µ ( x) ( x) c B µ B µ ( x) = min( µ ( x), µ ( x)) = µ ( x) ( x) c B µ B

(a) Fuzzy Sets and B (b) Fuzzy Set "not " B µ ( x) ( x) µ B.8.6 B.8.6 Is Contained in B.4.2.4.2 Membership Grades.8.6.4.2 B.8.6.4.2 (c) Fuzzy Set " OR B".8.6.4.2 (d) Fuzzy Set " ND B"

Not X X X X X

General requirements: Boundary: N()= and N() = Monotonicity: N(a) > N(b) if a < b Involution: N(N(a)) = a Two types of fuzzy complements: Sugeno s complement: N s( a ) = a + sa Yager s complement: w N ( a) = ( a ) / w w

Sugeno s complement: N s( a a ) = + sa Yager s complement: w N ( a) = ( a ) / w w N(a) (a) Sugeno's Complements s = -.95.8 s = -.7.6 s =.4 s = 2.2 s = 2.5 a N(a) (b) Yager's Complements w = 3.8 w =.5.6 w =.4 w =.7.2 w =.4.5 a

Generalized intersection (Triangular/T-norm, logical and) Basic requirements: Boundary: T(, a) = T(a,) =, T(a, ) = T(, a) = a Monotonicity: T(a, b) <= T(c, d) if a <= c and b <= d Commutativity: T(a, b) = T(b, a) ssociativity: T(a, T(b, c)) = T(T(a, b), c)

Generalized intersection (Triangular/T-norm) Examples: Minimum: lgebraic product: Bounded product: Drastic product: T ( a, b)=min(a, b) T (a, b)=a b T (a, b)=max(,( a+ b )) T (a, b)={ a if b= b if a= otherwise]

Minimum: Tm(a, b) lgebraic product: Ta(a, b) Bounded product: Tb(a, b) Drastic product: Td(a, b) (a) Min (b) lgebraic Product (c) Bounded Product (d) Drastic Product.5.5.5.5.5 b a.5.5 Y = b.5 X = a.5 Y = b.5 X = a.5 Y = b.5 X = a.5.5.5.5 2 Y = y X = x 2 2 Y = y X = x 2 2 Y = y X = x 2 2 Y = y X = x 2

Basic requirements: Boundary: S(, a) =, S(a, ) = S(, a) = a Monotonicity: S(a, b) < S(c, d) if a < c and b < d Commutativity: S(a, b) = S(b, a) ssociativity: S(a, S(b, c)) = S(S(a, b), c) Examples: Maximum: lgebraic sum: Bounded sum: Drastic sum S ( a,b)=max(a,b) S( a, b) = a + b a b S ( a,b)=min(,( a+b ))

Maximum: Sm(a, b) lgebraic sum: Sa(a, b) Bounded sum: Sb(a, b) Drastic sum: Sd(a, b) (a) Max (b) lgebraic Sum (c) Bounded Sum (d) Drastic Sum.5.5.5.5.5 Y = b.5 X = a.5 Y = b.5 X = a.5 Y = b.5 X = a.5 Y = b.5 X = a.5.5.5.5 2 Y = y X = x 2 2 Y = y X = x 2 2 Y = y X = x 2 2 Y = y X = x 2

Generalized De Morgan s Law T-norms and T-conorms are duals which support the generalization of DeMorgan s law: T(a, b) = N(S(N(a), N(b))) S(a, b) = N(T(N(a), N(b))) Tm(a, b) Ta(a, b) Tb(a, b) Td(a, b) Sm(a, b) Sa(a, b) Sb(a, b) Sd(a, b)