Fuzzy Sets. Mirko Navara navara/fl/fset printe.pdf February 28, 2019

Transcription

1 The notion of fuzzy set. Minimum about classical) sets Fuzzy ets Mirko Navara navara/fl/fset printe.pdf February 8, 09 To aviod problems of the set theory, we restrict ourselves to subsets of some universal set universe) X PX) denotes the set of all subsets of a set X A set A PX) is uniquely determined by its characteristic function indicator) µ A : X {0, }, { if x A, µ A x) = 0 if x A. Using the notation we may write Instead of µ ) A {}, we write µ A ), etc. In particular µ = 0, µ X =.. Definition of fuzzy sets A = {x X : µ A x) = } = {x X : µ A x) > 0}. µ A M) = {x X : µ Ax) M}, A = µ A ) ) {} = µ 0, ]. A fuzzy subset of a universe X a fuzzy set) is a mathematical object A described by its generalized) characteristic function membership function) µ A : X [0, ] Alternative notation: Ax) In this context, classical sets are called crisp or sharp. FX) denotes the set of all fuzzy subsets of a universe X Range: RangeA) = { α [0, ] : x X : µ A x) = α) } = µ A X) Height: upport: Core: ha) = sup RangeA) uppa) = { x X : µ A x) > 0 } = µ A corea) = { x X : µ A x) = } = µ A ) Examples of fuzzy sets A, B FR), A 0, ] ) 0 if x < 0, x if x [0, ], µ A x) = x if x, ], 0 if x >, µ B x) = if x = 3, if x = 4, 4 if x = 5, 0 otherwise. For finite fuzzy sets, we use an abbreviated notation like µ B = {3, ), 4, ), 5, 4 )}. Alternative notations: µ B = { /3, /4, 4 /5}, µ B = /3 + /4 + 4 /5.

2 3 ystem of cuts of a fuzzy set Definition: et A FX), α [0, ]. The α-level of A is the crisp set µ A α) = { x X : µ A x) = α }. The system of cuts of A is the mapping R A : [0, ] PX) which assigns to each α [0, ] the α-cut R A α) = µ ) { A [α, ] = x X : µa x) α }. The system of strong cuts is the mapping A : [0, ] PX), where A α) = µ ) { A α, ] = x X : µa x) > α}. Alternative notations of α-cuts: [A] α, [A] α, α A, α A RangeA) = { α [0, ] : µ A α) }, ha) = sup { α [0, ] : R A α) }, uppa) = A 0), corea) = R A ), R A 0) = X, A ) =. 4 The first representation theorem Theorem: A mapping M : [0, ] PX) is the system of cuts of some fuzzy set A FX) if and only if R) M0) = X, R) 0 α < β Mα) Mβ), R3) 0 < β Mβ) = Mα). α:α<β Proof: : R): M0) = R A 0) = X. R): x Mβ) = R A β) µ A x) β > α x R A α) = Mα). R3) : R) α [0, β) : Mβ) Mα) Mβ) Mα). α:α<β R3) : x α:α<β Mα) = α:α<β R A α) α [0, β) : µ A x) α, µ A x) β x R A β) = Mβ). : We shall prove that M = R A, where µ A x) := sup { α [0, ] : x Mα) }. : x Mβ) µ A x) β x R A β), : x R A β) µ A x) = sup { α [0, ] : x Mα) } β, α [0, β) : x Mα), x Mα) = Mβ). α:α<β 5 Representations of fuzzy set Horizontal representation: system of cuts Vertical representation: membership function Conversion from the horizontal to vertical representation: µ A x) = sup { α [0, ] : x R A α) }.

3 Theorem: the second representation theorem) et A FX). Then where the supremum is computed pointwise, i.e., 5. Fuzzy inclusion µ A = sup α µ RA α) = sup α µ RA α), α [0,] α RangeA) µ A x) = sup α µ RA α)x). α RangeA) Classical definition A B x A : x B cannot be used, because we cannot write x A, x B However, we can write A B x X : µ A x) µ B x) µ A µ B For A, B FX): A B x X : µ A x) µ B x) µ A µ B α [0, ] : R A α) R B α) Proof of the last equivalence: : Assume µ A µ B, x R A α), α µ A x) µ B x), x R B α), i.e., R A α) R B α) : Assume α [0, ] : R A α) R B α), µ A x) = sup { α [0, ] : x R A α) } sup { α [0, ] : x R B α) } = µ B x) 5. Cut-consistency A property P of fuzzy sets A,..., A n maps arguments A,..., A n to a truth value P A,..., A n ) {0, } predicate ). Property P of of fuzzy sets is called cutworthy if P A,..., A n ) α 0, ] : P R A α),..., R An α))), cut-consistent if P A,..., A n ) α 0, ] : P R A α),..., R An α))). 0-cuts are ignored intentionally) Examples: Inclusion is cut-consistent. trong normality, x X : µ A x) =, is cut-consistent. Crispness is cutworthy, but not cut-consistent. 6 Operations with fuzzy sets 6. Operations with crisp sets propositional set operations operations formula : PX) PX) : {0, } {0, } A = { x X : x A) } : PX) PX) : {0, } {0, } A B = { x X : x A) x B) } : PX) PX) : {0, } {0, } A B = { x X : x A) x B) } By means of membership functions: µ A x) = µ A x) µ A B x) = µ A x) µ B x) µ A B x) = µ A x) µ B x) 3

4 6. aws of Boolean algebras α = α, α β = β α, β = β α, α β) γ = α β γ), β) γ = β γ), β γ) = β) γ), α β γ) = α β) α γ), α α = α, α = α, α β) = α, α β) = α, α =, 0 = 0, α 0 = α, = α, α = 0, α α =, β) = α β, α β) = β. 6.3 Fuzzy negation unary operation. : [0, ] [0, ] such that α β. β. α, N) Example: tandard negation: α = α... α = α. N) Properties of fuzzy negations Theorem: Each fuzzy negation. is a continuous, strictly decreasing bijection satisfying. = 0,. 0 =. N0) Its graph is symmetric w.r.t. the axis of the st and 3rd quadrant, i.e.,. =. Proof: Injectivity: If. α =. β, then α =.. α =.. β = β. urjectivity: For each α [0, ] there is a β [0, ] such that α =. β, namely β =. α. continuity and boundary conditions. The symmetry of the graph is equivalent to involutivity N). Representation theorem for fuzzy negations A function. : [0, ] [0, ] is a fuzzy negation iff there is an increasing bijection i : [0, ] [0, ] generator of fuzzy negation. ) such that Proof: According to [Nguyen-Walker].) ufficiency: N): Assume α, β [0, ], α β. i, i preserve the ordering, reverses it:. = i i, i.e.,. α = i iα) ). iα) iβ) iα) iβ) i iα) ) i iβ) ). α. β N):.. = i i i i = i i = i i = id, where id is the identity on [0, ]. 4

5 Possible construction of a generator of a fuzzy negation Necessity: We shall prove that α +. α iα) = is a generator of a fuzzy negation.. i is increasing, continuous, and satisfies i0) = 0, i) =, thus i is a bijection on [0, ]. α +. α α +. α α +. α iα) = = = = α +. α.. α +. α = = = i. α). i =. i, i.e., i i =. A generator of a fuzzy negation is not unique. 6.4 Fuzzy complement µ A.x) =. µ A x). is the standard com- We distinguish them by the same indices as the corresponding fuzzy negations, e.g., A plement. 6.5 Fuzzy conjunction triangular norm, t-norm) binary operation. : [0, ] [0, ] such that, for all α, β, γ [0, ]:. β = β. α commutativity) T). β. γ) =. β). γ associativity) T) β γ. β. γ monotony) T3). = α boundary condition) T4) Theorem:. 0 = 0. Proof: Using T3) and T4):. 0 T3). 0 T4) = 0. Examples of fuzzy conjunctions tandard conjunction min, Gödel, Zadeh,... ): β = minα, β). ukasiewicz conjunction Giles, bold,... ): { α + β if α + β > 0, β = 0 otherwise. Product conjunction probabilistic, Goguen, algebraic product,... ): P β = α β. Drastic conjunction weak,... ): α if β =, D β = β if α =, 0 otherwise. 5

6 Properties of fuzzy conjunctions Theorem: α, β [0, ] : D β. β β. Proof: If α = or β =, then T4) gives the same result for all fuzzy conjunctions. Assume without loss of generality) that α β <. Then D β = 0. β. = α = β. Theorem: tandard conjunction is the only one which is idempotent, i.e., α [0, ] :. α = α Proof: Assume α, β [0, ], α β. thus. β = α = β. Analogously for α > β. α =. α T3). β T3). T4) = α, Representation of fuzzy conjunctions in general) Theorem: et be a fuzzy conjunction and i : [0, ] [0, ] be an increasing bijection. Then the operation : [0, ] [0, ] defined by β = i iα) iβ) ) is a fuzzy conjunction. If is continuous, so is. Proof: Commutativity analogously for associativity): Monotony: Assume β γ. Boundary condition: β = i iα) iβ)) = i iβ) iα)) = β α iβ) iγ), iα) iβ) iα) iγ), β = i iα) iβ)) i iα) iγ)) = γ. = i iα) i)) = i iα) ) = i iα)) = α. Classification of fuzzy conjunctions Continuous fuzzy conjunction. is Archimedean if strict if α 0, ) :. α < α TA) α 0, ] β, γ [0, ] : β < γ. β <. γ T3+) nilpotent if it is Archimedean and not strict. Example: Product conjunction is strict, ukasiewicz conjunction is nilpotent, standard and drastic conjunctions are not Archimedean the standard one violates TA), the drastic one is not continuous). 6

7 Representation theorem for strict fuzzy conjunctions Operation. : [0, ] [0, ] is a strict fuzzy conjunction iff there is an increasing bijection i : [0, ] [0, ] multiplicative generator) such that. β = i iα) P iβ) ) = i iα) iβ) ). ufficiency has been already proved except for strictness which is easy). The proof of necessity is much more advanced. A multiplicative generator of a strict fuzzy conjunction is not unique. Representation theorem for nilpotent fuzzy conjunctions Operation. : [0, ] [0, ] is a nilpotent fuzzy conjunction iff there is an increasing bijection i : [0, ] [0, ] ukasiewicz generator) such that. β = i iα) iβ) ). A ukasiewicz generator of a nilpotent fuzzy conjunction is not unique. Theorem: et. be a nilpotent fuzzy conjunction. Then n α 0, ) n N : k= α = 0. Proof: According to the representation theorem, it suffices without loss of generality) to prove the theorem for the ukasiewicz conjunction. For a sufficiently large n we obtain 6.6 Fuzzy intersection α + n α ) 0, is an operation on fuzzy sets defined using a fuzzy conjunction: i= µ A. Bx) = µ A x). µ B x) n k= α = 0. we distinguish them by the same indices as the respective fuzzy conjunctions) Theorem: The standard intersection is cut-consistent. Proof:. Cutworthiness: R A Bα) = {x X : µ A Bx) α} = {x X : µ A x) α) µ B x) α)} = {x X : µ A x) α} {x X : µ B x) α} = R A α) R B α). Cuts R A α) R B α) for all α 0, ]) determine a unique fuzzy set equal to A B. 6.7 Fuzzy disjunction triangular conorm, t-conorm) is a binary operation. : [0, ] [0, ] such that Theorem: α. =. Proof: α. 3) 0. 4) =. α. β = β. α α. β. γ) = α. β). γ β γ α. β α. γ α. 0 = α commutativity) ) associativity) ) monotony) 3) boundary condition) 4) 7

8 Examples of fuzzy disjunctions tandard max, Gödel, Zadeh... ): α β = maxα, β). ukasiewicz Giles, bold, bounded sum... ): { α + β for α + β <, α β = otherwise. Product probabilistic... ): Drastic weak... ): Einstein α P β = α + β α β. α for β = 0, α D β = β for α = 0, otherwise. α E β = α + β + αβ Properties of fuzzy disjunctions α, β [0, ] : α β α. β α D β. The standard disjunction is the only one which is idempotent, i.e., α. α = α for all α [0, ]. Duality et. be a fuzzy negation. to. ). to. ). A. If. is a fuzzy conjunction, then α. β =.... β) is a fuzzy disjunction dual to. with respect B. If. is a fuzzy disjunction, then. β =.. α.. β) is a fuzzy conjunction dual to. with respect Theorem: The ukasiewicz operations, are dual with respect to the standard negation. The product operations P, P are dual with respect to the standard negation. The standard operations, are dual with respect to any fuzzy negation. The drastic operations D, D are dual with respect to any fuzzy negation. Classification of fuzzy disjunctions A continuous fuzzy disjunction. is Archimedean if strict if nilpotent if it is Archimedean and not strict. α 0, ) : α. α > α A) α [0, ) β, γ [0, ] : β < γ α. β < α. γ 3+) 8

9 Representation theorems for fuzzy disjunctions Theorem: An operation. : [0, ] [0, ] is a strict fuzzy disjunction iff there is an increasing bijection i : [0, ] [0, ] such that α. β = i iα) P iβ) ). Theorem: An operation. : [0, ] [0, ] is a nilpotent fuzzy disjunction iff there is an increasing bijection i : [0, ] [0, ] additive generator) such that α. β = i iα) iβ) ) { i iα) + iβ) ) if iα) + iβ) = otherwise. 6.8 Fuzzy union is an operation on fuzzy sets defined using a fuzzy disjunction: µ A. B x) = µ A x). µ B x). we distinguish them by the same indices as the respective fuzzy disjunctions) Theorem: The standard union is cut-consistent. 6.9 Fuzzy propositional algebras equations written in black always hold equations written in red hold for the standard fuzzy operations, but not for some others equations written in blue hold only for some choices of fuzzy operations not for the standard ones).. α = α, α. β = β. α,. β = β. α, α. β). γ = α. β. γ),. β). γ =. β. γ),. β. γ) =. β).. γ), α. β. γ) = α. β). α. γ), α. α = α,. α = α, α.. β) = α,. α. β) = α, α. =,. 0 = 0, α. 0 = α,. = α,.. α = 0, α.. α =,.. β) =. α.. β,. α. β) =... β. 6.0 Fuzzy implication is any operation.. : [0, ] [0, ] which coincides with the classical implication on {0, }. We would like to satisfy the following properties, but we do not require them as axioms: α.. β = α β, α.. β = α β, Ia) Ib).. β = β, I).. is nonincreasing in the first argument and nondecreasing in the second, I3) α.. β = β.. α, α.. β.. γ) = β.. α.. γ), continuity. I4) I5) I6) 9

10 R-implication residuated fuzzy implication, residuum) is an operation α R. β = sup{γ :. γ β} RI) where. is a fuzzy conjunction if. is continuous, we may take the maximum instead of the supremum) Examples of R-implications From the standard conjunction we obtain the Gödel implication α R β = { if α β, β otherwise. It is piecewise linear and continous except for the points α, α), α <. From the ukasiewicz conjunction we obtain the ukasiewicz implication α R β = { if α β, α + β otherwise. It is piecewise linear and continous. From the product conjunction P we obtain the Goguen also Gaines) implication It has one point of discontinuity, 0, 0). Properties of R-implications α R P β = { if α β, β α otherwise. Theorem: et. be a continuous fuzzy conjunction. Then the R-implication R. Proof: α R. β = sup Γα, β), where satisfies Ia), Ib), I), I3). Γα, β) = {γ :. γ β} is an interval containing zero. Moreover, due to the continuity of. the interval is closed.) Ia) If α β, then Γα, β) = [0, ], sup Γα, β) =. Ib) If α > β, then / Γα, β), sup Γα, β) < from the closedness of Γα, β)). I): R. β = sup{γ : γ β} = β. I3): When α increases, Γα, β) does not increase. When β increases, Γα, β) does not decrease. Theorem: A residuated fuzzy implication induced by a continuous fuzzy conjunction. is continuous iff. is nilpotent. -implication is an operation α. β = α. β I) where. is a fuzzy disjunction Example: From the standard disjunction we obtain the Kleene Dienes implication α β = max α, β). 0

11 From the ukasiewicz disjunction we obtain the ukasiewicz implication which coincides with the ukasiewicz residuated implication R. Among all fuzzy implications studied here, only residuated implications induced by nilpotent fuzzy conjunctions e.g., the ukasiewicz implication) satisfy all properties Ia),Ib),I) I6). 6. Fuzzy biimplication equivalence) is an operation.. usually defined by α.. β = α.. β). β.. α), where.. is a fuzzy implication and. is a fuzzy conjunction biimplications are distinguished by the same indices as the respective fuzzy implications). If. satisfies Ia) e.g., for a residuated implication), at least one of the brackets equals, hence the choice of the fuzzy conjunction. is irrelevant. Example: ukasiewicz biimplication: α R β = α β. 7 Fuzzy relations 7. Classical relations A binary relation is an R X Y Inverse relation to R: R Y X: R = { y, x) Y X : x, y) R } The composition of relations R X Y, Y Z is R X Z: R = {x, z) X Z : y Y : x, y) R, y, z) )} Using membership functions: µ R : X Y {0, } µ R y, x) = µ R x, y) µ R x, z) = max µr x, y) µ y, z) ) y Y 7. Fuzzy relations A fuzzy relation is R FX Y ), µ R : X Y [0, ] The inverse relation to R is R FY X): x X y Y : µ R y, x) = µ R x, y) The -composition of relations R FX Y ), FY Z) is R. FX Z): µ R.x, z) = sup µr x, y). µ y, z) ) y Y Theorem The inversion of fuzzy relations is cut-consistent. Theorem If Y is a finite set, then the standard composition of fuzzy relations R FX Y ), FY Z) is cut-consistent.

12 7.3 pecial crisp relations R X X can be: an equality: E = { x, x) : x X }, reflexive: x X : x, x) R, i.e., E R, symmetric: x, y) R y, x) R, i.e., R = R, antisymmetric: x, y) R ) y, x) R ) x = y, i.e., R R E, transitive: x, y) R ) y, z) R ) x, z) R, i.e., R R R, a partial order: antisymmetric, reflexive, and transitive, an equivalence: symmetric, reflexive, and transitive. The membership function of the equality relation, E X X, is the Kronecker delta: { for x = y, µ E x, y) = δx, y) = 0 for x y. 7.4 pecial fuzzy relations A fuzzy relation R FX X) can be: reflexive: E R, symmetric: R = R, -antisymmetric: R. R E, -transitive: R. R R, a -partial order: -antisymmetric, reflexive, and -transitive, an -equivalence: symmetric, reflexive, and -transitive. The last four terms depend on the choice of the fuzzy conjunction.. Theorem The following properties of fuzzy relations are cut-consistent: reflexivity, symmetry, standard antisymmetry, standard transitivity, standard partial order, standard equivalence. 7.5 Projections of fuzzy relations The left first) projection of a fuzzy relation R FX Y ) is P R) FX): µ PR)x) = sup µ R x, y) y Y The right second) projection of a fuzzy relation R FX Y ) is P R) FY ): µ PR)y) = sup µ R x, y) x X Theorem The projections of fuzzy relations are cut-consistent. 7.6 Cylindric extension also the cartesian product) of fuzzy sets A FX), B FY ) is A B FX Y ): µ A B x, y) = µ A x) µ B y) It is the maximal fuzzy relation R FX Y ) such that P R) A and P R) B. Equality occurs iff ha) = hb). Theorem P R) P R) R Theorem The cylindric extension is cut-consistent.

13 8 Extension principle 8. The extension of binary relations to crisp sets A mapping is R X Y : x X!y = rx) Y : x, y) R A mapping R X Y corresponds to an r : X Y by x, y) R y = rx), R = { x, rx) ) : x X } The extension of a relation R X Y is a mapping r : PX) PY ): ra) = { y Y : x A : x, y) R )} Analogously, the extension of the relation R Y X is a mapping r : PY ) PX): r B) = { x X : y B : x, y) R )} The extensions r and r are mappings even if the original relation R was not a mapping. However, they are not mutually inverse. If, moreover, R is a mapping, then ra) = { rx) : x A } r B) = { x X : rx) B } In particular, Using membership functions: r y) = r {y}) = { x X : rx) = y } µ ra) y) = max µr x, y) µ A x) ) x X µ r B)x) = max µr x, y) µ B y) ) y Y 8. The extension of binary relations to fuzzy sets The extension of a relation R X Y is a mapping r : FX) FY ): µ ra) y) = sup µr x, y) µ A x) ) A FX), y Y ) x X Analogously, the extension of the relation R Y X is a mapping r : FY ) FX): µ r B)x) = sup µr x, y) µ B y) ) B FY ), x X) y Y As R is a crisp relation, the choice of the fuzzy conjunction. is irrelevant: { µ A x) for µ R x, y) = µ R x, y). µ A x) = 0 for µ R x, y) = 0 Using the extensions r : PX) PY ), r : PY ) PX) of relations R, R to crisp sets, the extensions to fuzzy sets can be written as If, moreover, R is a mapping, then If R is a mapping, then Theorem If the sets are finite for all y Y, then the equality holds. µ ra) y) = sup µ A x) x r y) µ r B)x) = sup µ B y) y rx) µ r B)x) = µ B rx)) µ ra) y) = µ A r y)) r R A α) ) R ra) α) r y) = {x X : x, y) R} 3

14 8.3 Convex fuzzy sets Here denotes a linear space. A crisp set A is called convex if Using membership functions: et X be a crisp convex subset of a linear space. A fuzzy set A FX) is called convex if x, y A λ 0, ) : λx + λ) y A min µ A x), µ A y) ) µ A λx + λ) y ) x, y X λ 0, ) : µ A λx + λ) y ) µa x) µ A y) Convexity of fuzzy sets has nothing in common with the convexity of its membership function! Theorem Convexity is cut-consistent. In particular, a fuzzy set of real numbers is convex iff all its cuts are intervals. 8.4 Fuzzy numbers and fuzzy intervals A fuzzy interval is an A FR) such that: upp A is a bounded set, For all α 0, ], the cut R A α) is a closed interval, R A ) i.e., R A ) is a nonempty closed interval). If, moreover, R A ) is a singleton, then A is called a fuzzy number. Fuzzy intervals are convex. The fuzzy interval inverse to a fuzzy interval A is A FR): µ A x) = µ A x) The extension principle for binary relations applied to the unary minus) R A α) = R A α) 8.5 Binary operations with fuzzy intervals {+,,, /} : R R can be understood as a crisp relation R R: µ y, z), x ) { for y z = x, = 0 otherwise. This can be extended by the already introduced) extension principle for binary relations to an operation FR ) FR); this has to be composed with the cylindric extension FR) FR) FR ). We obtain the binary operation : FR) FR) FR). A FR), B FR) A B FR R) A B = A B) FR) µ A B x) = µ A B) x) = sup µa B y, z) µ y, z), x )) y,z) R 4

15 = sup y,z) R,y = sup y,z) R,y µ A B y, z) z=x µa y) µ B z) ) z=x In particular, for = +: We compute the supremum of the function µ A y) µ B z) over all y R, z R such that y + z = x. This is the supremum of the function µ A x z) µ B z) over all z R because y + z = x y = x z). µ A+B x) = sup µa x z) µ B z) ), z R µ A B x) = sup µa x + z) µ B z) ), z R µ A B x) = sup µ A x/z) µ B z)), x 0, z R, z 0 µ A/B x) = sup µa x z) µ B z) ). z R Only for µ A B 0) we have to use the original definition because of problems with division by zero). In particular, for crisp intervals A = [a, b], B = [c, d] we obtain the interval arithmetic: The latter equality holds only for 0 [c, d]. [a, b] + [c, d] = [a + c, b + d], [a, b] [c, d] = [a d, b c], [a, b] [c, d] = [ minac, ad, bc, bd), maxac, ad, bc, bd) ], [a, b]/[c, d] = [ mina/c, a/d, b/c, b/d), maxa/c, a/d, b/c, b/d) ]. µ A B x) = max { µ A y) µ B z) : y, z R, y z = x }. In case of division we assume µ B 0) = 0.) Theorem The addition, subtraction, multiplication, and division of fuzzy intervals is cut-consistent. In case of division we assume zero membership to the divisor.) Theorem The addition, subtraction, and multiplication of fuzzy numbers resp. fuzzy intervals) is a fuzzy number resp. a fuzzy interval). The same holds for division unless zero is in the closure of the support of the divisor.) Any real number x R can be understood as a fuzzy number represented by a crisp singleton {x}); we denote it by x. Theorem Properties of operations with fuzzy intervals: 0 + A = A, 0 A = 0, A = A, A + B = B + A, A B = B A, A + B + C) = A + B) + C, A B C) = A B) C, A + B) = A B, A) B = A B) = A B), A) = A, A/B = A /B), µ A B+C) µ A B)+A C) In the latter inequality, equality occurs if A crisp number A = x). The following situations may happer for fuzzy intervals: A A 0, A + B) B A, A/A, A/B) B A, A B + C) A B + A C. 5