1998/12/21NTUST Ailab Li-we Pan1 Fuzzy object-oriented system design W. Pedrycz, Z.A. Sosnowski Fuzzy Sets and System (1998) p121~134 ref : 孫宗瀛、楊英魁, Fuzzy 控制：理論、實作與應用, 全華（民 83 ） p113 指導老師 ： 陳錫明教授 學生：潘立偉 學號： M 日期： 87/12/21

1998/12/21NTUST Ailab Li-we Pan2 index OO approach Linguistic variables as types in objects –Linguistic variables –Linguistic approximation Fuzzy inheritance classification in OO systems –Learning

1998/12/21NTUST Ailab Li-we Pan3 Fundamentals of OO approach Element size : Integer height :Integer Element Size Height (Element) (Element) Class Element { protected:// Attributes int size; int height; public:// Methods Element(int S; int H); int Size(); {return size;} int Height() {return height;) } Class Object

1998/12/21NTUST Ailab Li-we Pan4 Element size : Integer height:Integer Element Size Height Building x : Integer y :Integer Building location x location y Pedestrian Speed : Real accel. : Real Pedestrian speed acceleration Class Building : public Element { protected : int x; //Location int y; public : Building(int S, int H, int X, int Y) : Element(S, h); {x = X; y = Y;} void Location(); } class Pedestrian : public Element { protected : float speed; float acceleration; public : Pedestrian(int S,int H,float S,float A) : Element(S, H); {speed = s; acceleration = A; }; float Speed(); { return speed; } float Acceleration();{return acceleration;} }

1998/12/21NTUST Ailab Li-we Pan5 Static x : Integer y :Integer Static location Element size : Integer height:Integer Element Size Height Tree class Static {// Static scene element protected : int x; int y; public : Static(int X, int Y); {x = X; y = Y;} void Location(); } class Tree : public Element, public Static { public : Tree(int S, int H, int X, int Y) : Element(S, h), Static(X, Y); } Multiple inheritance

1998/12/21NTUST Ailab Li-we Pan6 Linguistic variables as types in objects Element size : Integer height :Integer Element Size Height class Element {// Attributes SIZE size; HEIGHT height; public : … }

1998/12/21NTUST Ailab Li-we Pan7 Linguistic variables T={t 1,t 2,…,t s } linguistic terms. {ex:Small, Medium, Large,...} H={h 1,h 2,…,h r } modifiers(hedges).{ex:more or less, very,...} linguistic variable L : –L =, –N : name of variable, –L(G) : family of labels if fuzzy sets defined in a given universe of discourse (the language generated by grammar G) –U : universe of discourse –G : syntactic rules –Sem : L(G) -> F(U)Semantics if the variable

1998/12/21NTUST Ailab Li-we Pan8 semantics : 3 components Membership function A i : linguistic term t i  T Linguistic modifiers h(A)(x) = A p (x)h(.) : linguistic modifier, p>0 –very A(x) = A²(x), more or less A(x) = A 0.5 (x) h(A)(x) = A(x  ) h(A(x)) = A p (x  ) linguistic expressions e 1 or e 2 or … or e k k = 1,2,…,s (s ≦ card(T)) e i : h i t j (i=1,2,…,r; j = 1,2,…,s) Sem(e 1 ore 2 or…ore k )=sem(e 1 ) ∪ sem(e 2 ) ∪... ∪ sem(e k )

1998/12/21NTUST Ailab Li-we Pan9 Syntax of the linguistic variable Context-free grammar G = V&  : set of terminal & nonterminal symbol P : production rules  : initial symbol –v = {t 1,t 2,…,t s,h 1,h 2,…,h r, or}. –  = { ,, } –P represented in the BNF notation

1998/12/21NTUST Ailab Li-we Pan10 –  ::= – ::= | or – ::= t i | m i t i |, i=1,2,…,r j = 1,2,…,s ex: –T = {small, medium, big} –H = {very, more_or_less} –some valid sentences very small small or more_or_less medium or very big etc.

1998/12/21NTUST Ailab Li-we Pan11 Linguistic approximation 1if A(x i ) ≦ B(x i ) B(x i )/A(x i )otherwise 1if A(x i ) ≦ B(x i ) 1-A(x i )+B(x i )otherwise 1-B(x i )+A(x i ) if A(x i ) ≦ B(x i ) 1-A(x i )+B(x i ) otherwise Method 1 : Method 2 : Method 3 : Method 4 : Poss(A,B)if Nec(b, A) > 0.5, Nec(B,A)/2 *Poss(B,A)Otherwise Selected definitions of similarity indexes ||.|| : similarity index

1998/12/21NTUST Ailab Li-we Pan12 Ex of linguistic approximation Triangular membership function U = [0…100] membership level 1/2 numerical interval (X) of width 2d V={A 1,A 2,A 3,A 4,A 5,very,more_or_less,or} –A i, i=1,2,…,r : generic terms represented as triangular fuzzy numbers –Sem(A i )={  i, m i,  i } –very A(x) = A²(x)more or less A(x) = A 0.5 (X) A1X A2A3 X()-d x() X()+d

1998/12/21NTUST Ailab Li-we Pan13 Fuzzy inheritance fuzzy inheritance cannot be explicitly defined at the object level and should be induced by linguistic approximation. Inheritance : add new attributes and methods or provide more specialized method. Must inheritance of attributes or be more precise. Inheritance hold at the level of classes,not objects. Keep the inheritance of attribute for fuzzy classes.

1998/12/21NTUST Ailab Li-we Pan14 (Cont.) Imposing some restriction on the set of possible values this attribute in the subclass can be assume. –L(G 1 )  L(G 2 ) L(G 1 ), L(G 2 ) the languages in the class & subclass

1998/12/21NTUST Ailab Li-we Pan15 Element size : SIZE[small, medium, large] height : HEIGHT[small, medium, large] Element Size Height Vehicle size : SIZE[medium, large] height : HEIGHT[small, medium] Vehicle Size Height Example derivation of classes - class vehicle & element

1998/12/21NTUST Ailab Li-we Pan16 Classification in OO system One profound applications of the hierarchies of class of object. Lower the level in the hierarchy, the more features become associated. The matching process applies to the corresponding attributes. class1 Object Top-down match similarity

1998/12/21NTUST Ailab Li-we Pan17 Ex of classification Collection if instances Instances LengthHeightSpeedAcceleration 1ShortShortFastHigh Long Low 4MediumMediumMediumMedium Fast ShortSlowLow 7ShortShortMediumMedium m shortv shortv fastv high 10very longm bigm mediumm low m - more or less;v - very

1998/12/21NTUST Ailab Li-we Pan18 M_vehicle length[short, medium, long] Speed[slow, medium, fast] height[small, medium, large] M_vehicle length[short, medium, long] Speed[slow, medium, fast] height[small, medium, large] M_vehicle length[short, medium, long] Speed[slow, medium, fast] height[small, medium, large] Hierarchy m_vehicle, m_car, m_truck