Generalized Interval-Valued Fuzzy Rough Sets Based on Interval-Valued Fuzzy Logical Operators

Transcription

1 nternational Journal of Fuzzy Systems Vol 5 o 4 Decemer Generalized nterval-valued Fuzzy Rough Sets Based on nterval-valued Fuzzy Logical Operators Bao Qing Hu and Heung Wong Astract he fuzzy rough sets and generalized fuzzy rough sets have een extended y three pairs of fuzzy logical operators to deal with real-valued data for a variety of models hree pairs of fuzzy logical operators are triangular norm or t-norm and its dual (t-conorm) residual implicator or R-implicator and its dual fuzzy implicator and t-norm which are frequently discussed in generalization models of rough sets here are recent researches into generalized interval-valued fuzzy rough sets for interval-valued fuzzy datasets However only fuzzy implicator and t-norm are used in generalized interval-valued fuzzy rough sets and another two pairs of fuzzy logical operators t-norm and t-conorm R-implicator and its dual have not een considered in generalized interval-valued fuzzy rough sets n this paper these models are generalized to a new approach that not only considers interval-valued fuzzy sets ut also two pairs of fuzzy logical operators First we study the interval representations of interval-valued fuzzy t-norm interval-valued fuzzy negator interval-valued fuzzy R-implicator and their duals which provide a theoretical asis for interval-valued fuzzy rough sets ased on interval-valued fuzzy logical operators Second we propose generalized interval-valued fuzzy rough sets ased on two pairs of fuzzy logical operators: interval-valued fuzzy t-norms and t-conorm and interval-valued fuzzy R-implicators and its dual Finally we confirm that some existing models including rough sets interval-valued fuzzy rough sets and generalized fuzzy rough sets ased on fuzzy logical operators are special cases of the proposed models Keywords: nterval-valued fuzzy sets interval-valued Corresponding Author: Bao Qing Hu is with the School of Mathematics and Statistics Wuhan University Wuhan P R China qhu@whueducn Heung Wong is with the Department of Applied Mathematics he Hong Kong Polytechnic University Hong Kong heungwong@polyueduhk Manuscript received 0 June 03; revised Aug 03; accepted Oct 03 fuzzy logical operators interval-valued fuzzy triangular norms interval-valued fuzzy residual implicators interval-valued fuzzy rough sets ntroduction he concept of the rough sets (RSs) was originally proposed y Pawlak [0] as a mathematical approach to study intelligent systems characterized y insufficient and incomplete information he usefulness and versatility of this theory have een amply demonstrated y successful applications in a variety of prolems such as knowledge discovery machine learning data analysis approximate classification conflict analysis and so forth t is well known that there are two asic elements in rough set theory crisp set and equivalence relation or indiscerniility relation which constitute the mathematical asis of RSs Crisp set and equivalence relation enale RSs to work well on some prolems ut also limit further applications of RSs For instance RS cannot work effectively on datasets with real values and interval values ecause there are errors due to the discretization of the data in rough sets models o overcome these prolems asic rough sets introduced y Pawlak are extended in relations ojects universes of discourse and operators etc Here are some examples of possile extensions ) Equivalence relations are extended to general relations fuzzy relations VF relations [0 6] type- fuzzy relations [9] and dominance relation [ 3] etc For example fuzzy rough sets (FRSs) [9 6] interval-valued fuzzy rough sets (VF-RSs) [ 3 7 3] generalized interval type- fuzzy rough sets [34] Dominance-ased rough sets ([5 7 3]) ) Ojects are extended to fuzzy sets VF sets and type- fuzzy sets for example rough fuzzy sets (RFSs [9]) interval-valued rough fuzzy sets (VF-RFSs [0]) VF-RSs [] and interval type- rough fuzzy sets [33] 3) Universes of discourse are extended from one to two such as generalized rough sets (GRSs) [5] generalized fuzzy rough sets (GFRSs) [9 5 6] generalized interval-valued fuzzy rough sets (GVF-RSs) [30] and generalized interval type- fuzzy rough sets [34] 4) Operators are extended from max and min to triangular norm (t-norm t-conorm) or fuzzy logical 03 FSA

2 38 nternational Journal of Fuzzy Systems Vol 5 o 4 Decemer 03 ( ) -GVF-RS new model proposed in this paper (VF relation VF set) ( )-GVF-RS new model proposed in this paper (VF relation VF set) ( ) -GVF-RS (VF relation VF set) VF rough set (VF relation VF set) ( ) -rough set (Fuzzy relation Fuzzy set) ( )-rough set (Fuzzy relation Fuzzy set) ( ) -rough set (Fuzzy relation Fuzzy set) Generalized fuzzy rough set (Fuzzy relation Fuzzy set) Fuzzy rough set (Fuzzy equivalence relation Fuzzy set) Rough fuzzy set (Equivalence relation Fuzzy set) Pawlak rough set (Equivalence relation Classical set) Figure he extension process of rough sets (in these parentheses the first item is relation; the second item is oject) operators for instance ( ) -GFRSs [4 8] ( ) -GFRSs [9] ( ) -GFRSs [ 4] ( is a fuzzy implicator []) and ( ) -GVF-RSs [30] t is worth noting that intuitionistic fuzzy rough sets were studied in [ ] which are equivalent to VF rough sets since intuitionistic fuzzy sets are equivalent to VF sets [8] n the following so we discuss VF rough sets For VF information systems there has only een few studies of VF-RSs [0 30] and some pairs of fuzzy logical operators (eg t-norm and t-conorm fuzzy residual operator and its dual) in VF-RSs have not een considered n fact it is valuale from oth theoretical and practical aspects to set up a framework that comines VF sets and fuzzy logical operators From the theoretical aspect the knowledge representation power of VF-RSs can e improved y setting up a roust framework so that further generalization of RSs is achieved From the practical aspect it is possile to propose new attriute reduction methods in an VF information system Motivated y such oservations we propose a hyrid model in this paper which generalizes the existing models RSs GFRSs and VF-RSs y considering not only VF sets ut also two fuzzy logical operators ie VF triangular norms and VF residual implicators We uild the theory of VF-RSs ased on these fuzzy logical operators Some existing models such as RSs GFRSs and VF-RSs are treated as special cases of the proposed models his theory provides a theoretical foundation for new attriute reduction in an VF information system Fig illustrates the relationships etween existing models and the proposed models and the extension process of rough set Fig shows the extension process of rough sets Equivalence relations on attriutes are extended to fuzzy equivalence relations fuzzy relations VF relations; classical sets on ojects are extended to fuzzy sets VF

3 B Q Hu and H Wong: Generalized nterval-valued Fuzzy Rough Sets Based on nterval-valued Fuzzy Logical Operators 383 sets; operators are extended to VF t-norm VF R-implicator VF implicator step y step he rest of this paper is organized as follows Section reviews the asic concepts aout rough sets fuzzy rough sets and generalized fuzzy rough sets n Section 3 we study VF t-norm VF t-conorm VF R-implicator and their duals Section 4 discusses generalized interval-valued fuzzy rough sets ased on VF triangle norms and VF residual implicator on two universes of discourse Before the end of Section 4 we give an illustrative example to illustrate our proposed models he last section concludes this paper Preliminaries n this section we review RSs FRSs and GFRSs ased on fuzzy logical operators which are necessary preliminaries for constructing the model of VF-RSs ased on fuzzy logical operators Let X and Y e two non-empty sets which are referred to as universes and F ( X ) e the set of all fuzzy sets on X and F ( X Y) e the set of all fuzzy relations from X to Y A fuzzy relation RF ( X Y) is referred to as a serial fuzzy relation from X to Y if for each x X there exists y Y such that Rxy ( ) R is a fuzzy equivalence relation on X if R is reflexive [ Rxx () x X ] symmetric [ Rxy ( ) Ryx () x y X ] and Rxz () sup Rxy ( ) Ryz () x yz X] transitive [ yx n this paper and are t-norm and t-conorm on [0 ] respectively A negator is a decreasing function :[0] [0] satisfying (0) and () 0 A negator is referred to as involutive if and only if x ( ( )) x for all x [0] Specially s ( x) x is an involutive negator Given a negator t-norm and t-conorm are called dual with regard to if ( x ( ) y ( )) ( ( xy )) n this paper we always assume that is an involutive negator and and are dual with respect to For two fuzzy sets A and B we define -union -intersection and complement pointwise y the following formulas ( A B)( x) A( x) B( x)( A B)( x) A( x) B( x) ( A )( x) ( A( x)) () he following are easily otained from the aove definitions Ai Ai Ai Ai () i i i i Let X and Y e two finite non-empty universes and R a fuzzy relation from X to Y hen the triple ( XYR ) is referred to as a generalized fuzzy approximation space in the following discussions he fuzzy approximation space was generalized to the generalized fuzzy approximation space y Wu and Zhang [6] he following definition is a more general one Definition ([4 8]): n the generalized fuzzy approximation space ( XYR ) for a fuzzy set AF the -lower and -upper approximation of A denoted respectively y R and R are fuzzy sets of X with the following memerships R ( x) inf ( R( x y)) A( y) yy R ( x) sup R( x y) A( y) (3) yy he pair of fuzzy sets ( R R ) is referred to as a ( ) -generalized fuzzy rough set ( ( ) -GFRS) Proposition ([4 8]): n the generalized fuzzy approximation space ( XYR ) for ( ) -GFRSs for any A BF the following properties hold: () R ( ) R X () if A B then R R and R R (3) ( R A ) ( R ) ( R A ) ( R ) (4) R ( AB) R R R ( A B) R R (5) R ( AB) R R R ( A B) R R (6) R is serial R R R ( ) R X We define R-implicator and its dual as follows: ( x y) sup [0]: ( x ) y ( x y) inf [0]: ( x ) y (4) t is easy to get the following equations from Eq (4) inf a sup ( a ) i i i sup ai inf ( ai ) (5) i i n the sequel will e a lower semi-continuous t-norm therefore its dual upper semi-continuous Let and e t-norm and t-conorm and dual with respect to involutive negator on [0 ] hen ( ( x y)) ( ( x) ( y)) (6) Definition : n the generalized fuzzy approximation space ( X YR ) for a fuzzy set AF the -lower and -upper approximation of A denoted respectively y R and R are two fuzzy sets of X with the following memerships i

4 384 nternational Journal of Fuzzy Systems Vol 5 o 4 Decemer 03 R ( x) inf R( x y) A( y) yy R ( x) sup ( R( x y)) A( y) (7) yy he pair of fuzzy sets ( R R ) is called a ( )-generalized fuzzy rough set ( ( )-GFRS) Mi and Zhang [9] first introduced ( )-GFRS for x ( ) x he following properties hold for ( )-GFRS Proposition : n the generalized fuzzy approximation space (X Y R) for approximation operators in Eq (7) the following properties hold for any two fuzzy sets A BF () R ( ) R X () A B R R and R R (3) R ( A ) ( R ) R ( A ) ( R ) (4) R ( A B) R R R( A B) R R (5) R ( A B) R R R( A B) R R (6) R is serial R R AF R ( ) R X 3 nterval-valued Fuzzy Logical Operator and nterval-valued Fuzzy Sets n order to consider fuzzy logical operators in GVF-RSs VF logical operators must e studied first hese VF logical operators are VF t-norm VF t-conorm VF negator VF R-implicator and their duals o this end the following notations are introduced for statement of convenience () [ a a ] 0a a f a [ a a] for any a [0] [ a a ] [ ] a a () [ ai a i ] i (any index set) then we define sup[ a a ] [sup a sup a ] i i i i i i i inf[ ai ai ] [inf ai inf ai ] (8) i i i Definition 3 (VF logical operators [6 30]): () An VF t-norm (An VF t-conorm ) is a () () () commutative associative mapping : () () () (resp : ) which is increasing in oth arguments and for which ([ a a ]) [ a a ] (resp ([ a a ]0) [ a a () ]) [ a a ] () An VF negator is a decreasing mapping () () : satisfying (0) and () 0 An VF negator is involutive if and only if () ( ([ a a ])) [ a a ] for all [ a a ] he VF negator s ([ a a ]) [ a a ] [ a a ] is usually referred to as the standard VF negator (3) Given an VF negator an VF t-norm and an VF t-conorm are called dual wrt if and only if they satisfy the following conditions: ([ a a ][ ]) ( ([ a a ]) ([ ])) (9) ([ a a ][ ]) ( ([ a a ]) ([ ])) () for all [ a a ][ ] (4) he VF R-implicator and its dual generated y an VF t-norm and an VF t-conorm are defined y respectively for any () [ a a ][ ] ; [ a a ][ ] () sup [ ] : ([ a a ][ ]) [ ] [ a a ][ ] () inf [ ] : ([ a a ][ ]) [ ] (0) Since intuitionistic fuzzy logical operators [5 7] are equivalent to VF logical operators in the following we use VF logical operators Lemma 3 ([7 30]): Given two t-norms and and two t-conorms and satisfying and then an VF t-norm and an VF t-conorm are constructed y the following Eq () and Eq () () for two intervals [ a a ][ ] ([ a a ][ ]) [ ( a ) ( a )] () ([ a a ][ ]) [ ( a ) ( a )] () An VF t-norm (resp VF t-conorm ) is called t-representale (resp s-representale) if it can e represented in the form of Eq () (resp Eq ()) n the following discussion we always suppose that VF t-norm and VF t-conorm are t-representale and s-representale respectively and they are written as =[] and [ ] respectively Specially () [ a a ][ ] [ a a ] [ ] [ a a ] [ a a ] [ ] [ a a ] Lemma 3: Given a (an involutive) negator an (involutive) VF negator is constructed for any () [ a a ] y using Eq (3) elow ([ a a ]) [ ( a ) ( a )] (3) Proposition 3: VF t-norm =[ ] and VF t-conorm [ ] e dual wrt if and only if

5 B Q Hu and H Wong: Generalized nterval-valued Fuzzy Rough Sets Based on nterval-valued Fuzzy Logical Operators 385 t-norm and t-conorm e dual wrt and t-norm and t-conorm e dual wrt Proof: Let VF t-norm =[ ] and VF t-conorm () [ ] e dual wrt hen [ a a ][ ] we have [ a a ] = [ a a ][ ] ( ([ a a ]) ([ ])) ( ( a ) ( )) ( ( a ) ( )) hus ( a ) ( ( a ) ( )) ( a )= ( ( a ) ( )) amely t-norm and t-conorm e dual wrt and t-norm and t-conorm e dual wrt And vice versa Proposition 3: Let VF t-norm =[ ] and VF t-conorm [ ] e dual wrt hen () ([ a a ][ ]) () ([ a a ] [ ]) ( a ) ( a ) ( a ) (3) ([ a a ][ ]) ( ([ a a ]) ([ ])) Proposition 33: Let and e an VF t-norm and an VF t-conorm respectively hen () [ a a ][ ][ c c ] () [ a a ] [ a a ] 0[ a a ] [ a a ] () [ a a ] [ ] [ a a ][ c c ] [ ][ c c ] ( a ) ( a ) ( a ) [ c c ][ a a ] [ c c ][ ] a a c c c c [ c c ][ a a ] [ c c ][ ] [ ][ ] [ ][ ] Proposition 34: Let =[ ] and [ ] e an VF t-norm and an VF t-conorm respectively and and e continuous hen () [ a a ][ ][ c c ] () [ a a ] [ a a ][ ] [ a a ] [ a a ][ ] () [ a a ][ ] [ c c ] [ a a ][ c c ] [ ] [ a a ][ ] [ c c ] [ a a ] [ a a ][ ] [ ] [ a a ] [ a a ][ ] [ ] [ a a ][ c c ] [ ] (3) [ a a ][ ] [ c c ] [ a a ] [ ][ c c ] [ ] [ a a ][ c c ] [ a a ][ ] [ c c ] [ a a ] [ ][ c c ] [ ] [ a a ][ c c ] (4) [ a a ][ ] [ ][ c c ] [ a a ][ c c ] [ a a ][ ] [ ][ c c ] [ a a ][ c c ] (5) [ a a ][ ] [ ] [ a a ] [ a a ][ ] [ ] [ a a ] (6) [ a a ] [ ] [ a a ][ ] a a a a (7) [ a a ] [ ][ a a ] [ a a ] [ ][ a a ] 0 (8) [ a a ][ ] [ ] [ a a ][ ] [ ] [ ] [ ] [ ][ ] 0 Proposition 35: Let =[ ] and [ ] e an VF t-norm and an VF t-conorm wrt VF involutive () negator on respectively and and () e continuous on [0] and [ a a ] i [ ] () hen () sup[ ai ai ][ ] inf [ ai ai ][ ] i i inf[ a a ][ ] sup [ a a ][ ] i i i i i i i () [ ]inf[ ] a i a i inf [ ][ a i a i ] i i [ ]sup[ ai ai ] sup [ ][ ai ai ] i i (3) inf[ a ][ ] i a i sup [ a i a i ][ ] i i sup[ ai ai ][ ] inf [ ai ai ][ ] i i (4) [ ]sup[ ai ai ] sup [ ][ ai ai ] i i [ ]inf[ a a ] inf [ ][ a a ] i i i i i i n the following discussion let us always assume that =[ ] and [ ] are an VF t-norm and an i

6 386 nternational Journal of Fuzzy Systems Vol 5 o 4 Decemer 03 VF t-conorm wrt VF involutive negator on () respectively and and are continuous on [0] () A mapping A: X is called an VF set of X and A( x ) is its memership function [9] F () ( X ) is a set () of all VF sets of X For [ ] and M X we define M [ ] F () ( X ) with the following memership function: [ ] x M M[ ]( x) 0 x M M[ ] is also denoted y [ ] M For example F ( ) with the following memership function { x} () X { x} ( y) when y x otherwise 0 f M {} x then {} x [ ] is tersely denoted y x[ ] For any A [ A A ] B[ B B ] F () ( X) we define -union and -intersection pointwise y the formulas ( A B)() x Ax () B() x [ A () x B () x A () x B ()] x ( A B)() x Ax () Bx () [ A () xb () x A () x B ()] x (4) ( A )( x) ( Ax ( )) [ ( A ( x)) ( A ( x))] Specially ( A B)( x) [ A ( x) B ( x) A ( x) B ( x)] ( A B)( x) [ A ( x) B ( x) A ( x) B ( x)] ( s A )( x) A( x) [ A ( x) A ( x)] he order relation in F () ( X ) is defined y A B if and only if A() x B() x if and only if A () x B () x and A () x B () x for all x X wo sets and X are special elements in F () ( X ) with memership functions () x 0 and X() x x X respectively R F () ( X Y) is called an VF relation from X to Y Definition 3: For an VF relation R F () ( X X) () R is reflexive if Rxx ( ) for any x X () R is symmetric if Rxy ( ) Ryx () for any x y X (3) R is -transitive if Rxz ( ) ( Rxy ( ) Ryz ( )) for any x yz X (4) R is serial if x X y Y such that R x y An VF relation R on X is referred to as an VF -equivalence relation on X if it is reflexive symmetric and -transitive t follows from Proposition 34() that R is -transitive if and only if Ryz ( ) ( Rxy ( ) Rxz ( )) for any x yz X 4 Generalized nterval-valued Fuzzy Rough Sets on wo Universes of Discourse n the following we consider not only VF sets ut also VF logical operators in generalized rough sets Based on VF logical operators studied in the aove section we discuss generalized VF rough sets ased on triangular norms and residual implicators 4 Generalized nterval-valued Fuzzy Rough Sets Based on riangular orms Let X and Y e two finite universes and R an VF relation from X to Y hen the triple ( X YR ) is referred to as a generalized VF approximation space Definition 4: n the generalized VF approximation space ( X YR ) for an VF set A F () ( Y ) the generalized VF lower approximation R and the generalized VF upper approximation R of A are respectively defined as follows For any x X R ( x) inf ( R( x y)) A( y) yy (5) R ( x) sup R( x y) A( y) yy he pair of VF sets ( R R ) is referred to as a ( ) -generalized VF rough set ( ( ) -GVF-RS) Clearly the aove definition implies the following equivalence forms of Eq (5): x X R ( x) inf ( R( x y)) A( y) yy R xy A y R xy A y inf ( ( )) ( ) inf ( ( )) ( ) yy yy R ( A )( x) R ( A )( x) R ( A ) R ( A ) ( x) (6) R ( x) sup R( x y) A( y) yy A y R x y A y R x y sup ( ) ( ) sup ( ) ( ) yy yy R ( A )( x) R ( A )( x) R ( A ) R ( A ) ( x) (7) f an VF relation and an VF set are degenerated into a fuzzy relation and a fuzzy set respectively then a ( ) -GVF-RS is changed to a ( ) -GFRS Sun Gong Chen [] discussed ( ) -GVF-RS for a special case on = and ( x) x Proposition 4: n the generalized VF approximation space ( XYR ) for approximation operators in Eq (5) there are the following properties for any A B F () () R ( ) R X () A BR R and R R

7 B Q Hu and H Wong: Generalized nterval-valued Fuzzy Rough Sets Based on nterval-valued Fuzzy Logical Operators 387 (3) ( R A ) ( R ) R ( A ) ( R ) (4) R( AB) R R R( A B) R R (5) R( AB) R R R( A B) R R (6) R [ ] R ( A [ ] ) X R [ ] X R( A [ ] Y ) (7) R is serial R R R AF ( Y ) ( ) R X () Y () [ ] Proof: () t follows from a a [ a a ] [ ]0 0 and () t follows from the monotonicity of and (3) t is follows from the duality of and wrt (4) R ( A B) R ( A B) R ( A B) R A B R A B R ( A ) R ( B ) R ( A ) R ( B ) R ( A ) R ( A ) R ( B ) R ( B ) R R R( AB) R ( AB) R ( AB) R A B R A B R ( A ) R ( B ) R ( A ) R ( B ) R ( A ) R ( A ) R ( B ) R ( B ) R R (5) t is straightforward from the monotonicity in item () (6) For any () [ ] R [ ] X ( x) R( x)[ ] inf Ay ( ) Rxy ( ( )) [ ] yy inf Ay ( ) Rxy ( ( )) [ ] yy inf A( y)[ ] ( R( x y)) yy R( A [ ] Y ) he second proposition can e similarly proven (7) t can e proved in a similar way for Proposition (6) since an VF relation R [ R R ] is serial if and only if R and R are serial 4 Generalized nterval-valued Fuzzy Rough Sets Based on Residual mplicators Let us consider VF residual implicators in generalized VF rough sets as follows Definition 4: n the generalized VF approximation space ( XYR ) for an VF set AF () ( Y ) the generalized VF lower and generalized VF upper approximation of A denoted respectively y R and R are two VFSs of X with the following memerships R ( x) inf R( x y) A( y) yy (8) R ( x) sup ( R( x y)) A( y) yy he pair of VF sets ( R R ) is called a ( )-generalized VF rough set ( ( )- GVF-RS) Clearly the aove definition implies the following equivalence forms of Eq (8): x X R ( x) inf R( x y) A( y) yy R x y A y R x y A y yy R x y A y inf ( ) ( ) inf ( ) ( ) yy inf ( ) ( ) yy R ( A )( x) R ( A )( x) R ( A )( x) R ( A ) R ( A ) R ( A ) ( x) (9) R ( x) sup ( R( x y)) A( y) yy sup ( R ( x y)) A ( y) yy sup R ( ( xy )) A ( y) sup R ( ( xy )) A ( y) yy yy R ( A )( x) R ( A )( x) R ( A )( x) R ( A ) R ( A ) R ( A ) ( x) (0) f an VF relation and an VF set are degenerated into a fuzzy relation and a fuzzy set respectively then ( )-GVF-RS is changed to ( )-GFRS Proposition 4: n the generalized VF approximation space ( X YR ) for approximation operators in Eq (8) there are the following properties for any A B F () () R ( ) R X () A B R R and R R (3) R ( A ) ( R ) ( R A ) ( R ) (4) R ( A B) R R R( A B) R R (5) R ( A B) R R R( A B) R R (6) R is serial R R AF () ( Y )

8 388 nternational Journal of Fuzzy Systems Vol 5 o 4 Decemer 03 R ( ) R X Proof: Proofs of () () and (3) follow from Eqs (8) (9) and (0) (4) R ( A B) R (( A B) ) R (( A B) ) R (( A B) ) R ( A B ) R ( A B ) R ( A B ) R A R B R A R B ( ) ( ) ( ) ( ) R ( A ) R ( B ) R ( A ) R ( A ) R ( A ) R ( B ) R ( B ) R ( B ) R A B R A R B R A R B So ( ) ( ) ( ) he second part can e proved in a similar way (5) t is straightforward from the monotonicity in item () (6) t is proved similar to Proposition (6) ecause an VF relation R [ R R ] is serial iff R and R are serial 43 An Example Before the end of our work in order to understand the generalized interval-valued fuzzy rough sets ased on triangular norms and residual implicators an example is given as follows Let X Y { x x x6} =[ ] with (ordinary multiplication) and =[ ] with and ˆ ( ˆ aa a ) s ( ( x) x) hen s a a ( a ) ( a ) a a a 0 a 0 a ( a ) ( a ) a a a a An interval-valued fuzzy relation R is shown in ale An VF set A and its the VF lower approximations R R and upper approximatiosn R R are shown in ale where R and R are computed y Eq(6) and Eq(7) respectively and R and R are computed y Eq(9) and Eq(0) respectively rough sets ased on interval-valued fuzzy triangular norms and interval-valued fuzzy residual implicators have een studied We have considered generalizing the rough set model y synthesizing the following four aspects () Universes of discourse are two universes X and Y (may e different) () nformation system is an interval-valued fuzzy information ie R is an interval-valued fuzzy relation from X to Y (3) Oject is an interval-valued fuzzy set on the second universe Y (4) nterval-valued t-norm and t-conorm and interval-valued fuzzy residual implicator ased on t-norm and ased on t-conorm are considered he proposed hyrid model considering these four aspects provides a theoretical asis for a wide range of applications in rough set theory A large numer of rough set models are special examples of our new models We can see that different models are uilt upon four different aspects ie universe relation oject and operators he application domain of our new models will e the roadest in the aove four aspects he contained relations among models are shown in Fig Existing models and new models proposed in this paper are indicated y solid lines and dashed lines respectively his paper supports a theory to deal with interval-valued fuzzy datasets although the algorithms of attriute reductions ased on it have not een discussed in this paper his theory is a foundation of another paper in which variale precision prolem [ ] will e considered he discerniility matrix approach which is to investigate the structure of attriute reductions in interval-valued fuzzy variale precision rough sets ased on fuzzy operators and the algorithms to find all reductions will e studied in another paper t deals with the applications of interval-valued fuzzy variale precision rough sets along with experimental comparisons to show the feasiility and effectiveness of interval-valued fuzzy variale precision rough sets Acknowledgment his work was jointly supported y grants from he Hong Kong Polytechnic University Research Committee and he ational atural Science Foundation of China (Grant o ) 5 Conclusion n this paper the generalized interval-valued fuzzy

9 B Q Hu and H Wong: Generalized nterval-valued Fuzzy Rough Sets Based on nterval-valued Fuzzy Logical Operators 389 ale VF relation R R x x x 3 x 4 x 5 x 6 x [00] [0305] [00] [050] [0709] x [00] [04506] [0809] [07508] [0050] x 3 [0305] [04506] [04507] [055065] [03045] x 4 [00] [0809] [04507] [03045] [0809] x 5 [050] [07508] [055065] [03045] [0507] x 6 [0709] [0050] [03045] [0809] [0507] ale VF set A and its the VF lower approximations R R and upper approximations R R R A R R R R x [050] [050] [055085] [050] [ ] x [07085] [0036] [07085] [050] [07085] x 3 [0506] [0306] [0506] [0 0] [ ] x 4 [00] [00] [06085] [00] [075085] x 5 [04055] [04055] [05508] [0 0] [07508] x 6 [075085] [0036] [075085] [050] [075085] ( )-GFRS ( )-GVFF-RS VF RS GFRS FRS RFS Pawlak RS ( ) -GFRS ( ) -GVF-RS ( ) -GVFF-RS ( ) -GFRS Figure Relation diagram of existing models (solid) and new models (dashed) References [] M Baczyński and B Jayaram Fuzzy mplications Springer-Verlag Berlin Heidelerg 008 [] Y Cheng and D Miao Rule extraction ased on granulation order in interval-valued fuzzy information system Expert Syst Appl vol 38 pp [3] Y Cheng D Miao and Q Feng Positive approximation and converse approximation in interval-valued fuzzy rough sets nf Sci vol 8 no pp

10 390 nternational Journal of Fuzzy Systems Vol 5 o 4 Decemer 03 [4] C Cornelis M De Cock and E E Kerre ntuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge Expert Syst vol 0 no 5 pp [5] C Cornelis G Deschrijver and E E Kerre mplication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction classification application nt J Approx Reason vol 35 no pp [6] C Cornelis G Deschrijver and E E Kerre Advances and challenges in interval-valued fuzzy logic Fuzzy Sets Syst vol 57 pp [7] G Deschrijver C Cornelis and E E Kerre On the representation of intuitionistic fuzzy t-norms and t-conorms EEE rans on Fuzzy Syst vol no pp [8] G Deschrijver and E E Kerre On the position of intuitionistic fuzzy set theory in the framework of theories modeling imprecision nf Sci vol 77 pp [9] D Duios and H Prade Rough fuzzy sets and fuzzy rough sets nt J Gen Syst vol 7 pp [0] Z Gong B Sun and D Chen Rough set theory for the interval-valued fuzzy information systems nf Sci vol 78 pp [] S Greco M nuiguchi and R Slowinski Fuzzy rough sets and multiple-premise gradual decision rules nt J Approx Reason vol 4 no pp [] S Greco B Matarazzo and R Słowiński Rough sets theory for multicriteria decision analysis Eur J Oper Res vol 9 pp [3] S Greco B Matarazzo and R Słowiński Rough approximation y dominance relations nt J ntell Syst vol 7 pp [4] B Q Hu and Z H Huang ( )-Generalized fuzzy rough sets ased on fuzzy composition operations in: B-y Cao C-y Zhang and -f Li (Eds): Fuzzy nformation and Engineering ASC 54 Springer-Verlag Berlin Heidelerg pp [5] B Huang D-K Wei H-X Li and Y-L Zhuang Using a rough set model to extract rules in dominance-ased interval-valued intuitionistic fuzzy information systems nf Sci vol pp [6] A De Korvin C McKeegan and R Kleyle Knowledge acquisition using rough sets when memership values are fuzzy sets J ntell Fuzzy Syst vol 6 pp [7] Y Leung M M Fischer W-Z Wu and J-S Mi A rough set approach for the discovery of classification rules in interval-valued information systems nt J Approx Reason vol 47 pp [8] J-S Mi Y Leung H-Y Zhao and Feng Generalized fuzzy rough sets determined y a triangular norm nf Sci vol 78 pp [9] J-S Mi and W-X Zhang An axiomatic characterization of a fuzzy generalization of rough sets nf Sci vol 60 pp [0] Z Pawlak Rough sets nt J Comput nf Sci vol pp [] A M Radzikowska and E E Kerre A comparative study of fuzzy rough sets Fuzzy Sets Syst vol 6 pp [] B Sun Z Gong and D Chen Fuzzy rough set theory for the interval-valued fuzzy information systems nf Sci vol 78 pp [3] H Wu Y Wu and J Luo An interval type- fuzzy rough set model for attriute reduction EEE rans on Fuzzy Syst vol 7 no pp [4] W-Z Wu Y Leung and J-S Mi On characterizations of ( )-fuzzy rough approximation operators Fuzzy Sets Syst vol 54 pp [5] W-Z Wu J-S Mi and W-X Zhang Generalized fuzzy rough sets nf Sci vol 5 pp [6] W-Z Wu and W-X Zhang Constructive and axiomatic approaches of fuzzy approximation operators nf Sci vol 59 pp [7] X Yang D Yu J Yang and L Wei Dominance-ased rough set approach to incomplete interval-valued information system Data Knowl Eng vol 68 pp [8] Y Y Yao S K M Wong and P Lingras A decision-theoretic rough set model Proceeding of the 5 th nt Symposium on Methodologies for ntelligent Syst 990 [9] L A Zadeh he concept of a linguistic variale and its applications to approximate reasoning nf Sci vol 8 pp ; vol 8 pp ; vol 9 pp [30] H-Y Zhang W-X Zhang and W-Z Wu On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse nt J of Approx Reason vol 5 pp [3] H-Y Zhang Y Leung and L Zhou Variale-precision-dominance-ased rough set approach to interval-valued information systems nf Sci vol 44 pp [3] X Zhang B Zhou and P Li A general frame for intuitionistic fuzzy rough sets nf Sci vol 6 pp

11 B Q Hu and H Wong: Generalized nterval-valued Fuzzy Rough Sets Based on nterval-valued Fuzzy Logical Operators 39 [33] Z Zhang On interval type- rough fuzzy sets Knowledge-Based Syst vol 35 pp -3 0 [34] Z Zhang On characterization of generalized interval type- fuzzy rough sets nf Sci vol 9 pp [35] S Zhao E C C sang and D Chen he model of fuzzy variale precision rough sets EEE rans on Fuzzy Syst vol 7 pp [36] L Zhou and W-Z Wu On generalized intuitionistic fuzzy rough approximation operators nf Sci vol 78 pp [37] Lei Zhou W-Z Wu and W-X Zhang On characterization of intuitionistic fuzzy rough sets ased on intuitionistic fuzzy implicators nf Sci vol 79 pp [38] W Ziarko Variale precision rough set model J Comput Syst Sci vol 46 pp Bao Qing Hu is a Professor at the School of Mathematics and Statistics Wuhan University (China) He otained his BSc MSc and PhD from Wuhan University His current research interests include fuzzy mathematics rough set theory and soft computing Heung Wong is a Professor at the Department of Applied Mathematics the Hong Kong Polytechnic University He otained his BSc (Mathematics) MSc (Statistics) and PhD (Statistics) from he Chinese University of Hong Kong ewcastle University (UK) and he University of Hong Kong respectively His current research interests include time series analysis environmental statistics and stochastic hydrology