Relational Interpretations of Neighborhood Operators and Rough Set Approximation Operators

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1 Relatioal Iterpretatios of Neighborhood Operators ad Rough Set Approximatio Operators Y.Y. Yao Departmet of Computer Sciece, Lakehead Uiversity, Thuder Bay, Otario, Caada P7B 5E1, This paper presets a framework for the formulatio, iterpretatio, ad compariso of eighborhood systems ad rough set approximatios usig the more familiar otio of biary relatios. A special class of eighborhood systems, called 1-eighborhood systems, is itroduced. Three extesios of Pawlak approximatio operators are aalyzed. Properties of eighborhood ad approximatio operators are studied, ad their coectios are examied. Key words: Approximatio operators, biary relatios, eighborhood systems, rough sets, partitios, coverigs. 1 INTRODUCTION The theory of rough sets is motivated by practical eeds i classificatio, cocept formatio, ad data aalysis with isufficiet ad icomplete iformatio [12 15]. It provides a systematic approach for the study of idisceribility of objects. Typically, idisceribility is described usig equivalece relatios. Whe objects of a uiverse are represeted by usig a set of attributes, oe may defie the idisceribility of objects based o their attribute values. If two objects are characterized by the same values o certai attributes, i.e., they have the same descriptio, they are said to be idistiguishable or equivalet. All objects with the same descriptio form a equivalece class. The family of equivalece classes defies a partitio of the uiverse. I terms of equivalece classes, a subset of the uiverse may be approximated by two subsets. The lower approximatio is the uio of equivalece classes which are subsets of the give set, ad the upper approximatio is the uio of equivalece classes which have a oempty itersectio with the give set. They ca be formally described by a pair of uary set-theoretic operators [23]. By applyig the argumet i a wider cotext, oe may geeralize the otio of approximatio Preprit submitted to Elsevier Preprit 2 December 2007

2 operators by usig o-equivalece relatios [10,19,20,22,24], or a coverig of the uiverse [10,16,25]. This leads to various approximatio operators. By adoptig the otio of eighborhood systems from topological space ad its geeralizatio called Frechet (V)space [18], Li [4,5,7,8] proposed a more geeral framework for the study of approximatio. I eighborhood systems, each elemet of a uiverse is associated with a oempty family of subsets of the uiverse. This family is called a eighborhood system of the elemet, ad each member is called a eighborhood of the elemet. A subset of the uiverse ca be approximated based o eighborhood systems of all elemets i the uiverse. With respect to a equivalece relatio, the equivalece class cotaiig a give elemet may be iterpreted as a eighborhood of that elemet [10,20]. For a arbitrary biary relatio, the successor elemets of a give elemet may be iterpreted as its eighborhood [24]. The theory of rough sets built from biary relatios may therefore be related to eighborhood systems. The mai objective of this paper is to build a commo framework for the study of a special class of eighborhood systems ad rough set approximatios. Biary relatios are used as a primitive otio to iterpret various cocepts ivolved. Withi the proposed framework, mai results of studies o rough set approximatio operators are reviewed. Pawlak approximatio operators are exteded i three directios. Oe extesio is cosistet with the iterpretatio of ecessity ad possibility operators of modal logic. The resultig approximatio operators ca be expressed i terms of eighborhood operators. The other two methods use coverigs of uiverse iduced by eighborhood systems. The coectios betwee biary relatios, eighborhood operators, ad approximatio operators are examied. 2 RELATION BASED NEIGHBORHOOD OPERATORS Let U deote a fiite ad oempty set called the uiverse. For each elemet x of U, oe associates it with a subset (x) U called a eighborhood of x. A eighborhood of x may or may ot cotai x. A eighborhood system NS(x) of x is a oempty family of eighborhoods of x. A eighborhood system of U, deoted by NS(U), is the collectio of NS(x) for all x i U. It determies a Frechet (V)space, writte (U, NS(U)). There is o additioal requiremet o the eighborhood systems i a Frechet (V)space. A topological space is a Frechet (V)space, but the coverse is ot true [5,6]. I this study, we cosider a special type of eighborhood systems i which each elemet has exactly oe eighborhood. They are called 1-eighborhood systems. Such a eighborhood system ca be described by a eighborhood 2

3 operator : U 2 U, where 2 U deotes the power set of the uiverse. The operator assigs a uique eighborhood (x) to each elemet x U. For a fiite uiverse U, we ca exted a eighborhood operator from U to 2 U as follows: N(X) = x X (x). (1) For the empty set, we defie N( ) =. The mappig N : 2 U 2 U associates each subset of U with a subset of U. It may be cosidered as a uary set-theoretic operator. By defiitio, we have N({x}) = (x). Operator N may therefore be iterpreted as a additive extesio of. I order to study the structures of 1-eighborhood systems, we cosider the followig properties of a eighborhood operator: serial : for all x U, there exists a y U such that y (x), for all x U, (x), iverse serial : for all x U, there exists a y U such that x (y), (x) = U, x U reflexive : for all x U, x (x), symmetric : for all x, y U, x (y) = y (x), trasitive : for all x, y, z U, [y (x), z (y)] = z (x), Euclidea : for all x, y, z U, [y (x), z (x)] = z (y). A reflexive eighborhood operator is both serial ad iverse serial. The family of eighborhoods {(x) x U} of a iverse serial eighborhood operator forms a coverig of the uiverse. By combiig these properties, we ca characterize more classes of eighborhood systems [8]. A eighborhood system is called a B-eighborhood system if the eighborhood operator is reflexive ad symmetric. It is called a Pawlak-eighborhood system if the eighborhood operator is reflexive, symmetric, ad trasitive [5,8]. A Pawlak-eighborhood system ca be equivaletly characterized by the properties of reflexivity ad Euclidea. The family of eighborhoods of a Pawlak eighborhood operator forms a partitio of the uiverse. The class of 1-eighborhood systems ca be iterpreted usig the more familiar otio of biary relatios. A biary relatio R over a uiverse U is a subset of the Cartesia product U U. For two elemets x, y U, if xry, we say that y is R-related to x, x is a predecessor of y, ad y is a successor of x. Give a biary relatio, we defie the successor eighborhood of x as follows: R s (x) = {y xry}. (2) 3

4 It ca be viewed as a successor eighborhood operator from U to 2 U. Properties of a biary relatio R ca be stated usig the successor eighborhood operator: serial : for all x U, there exists a y U such that xry, for all x U, R s (x), iverse serial : for all x U, there exists a y U such that yrx, R s (x) = U, x U reflexive : for all x U, xrx, for all x U, x R s (x), symmetric : for all x, y U, xry = yrx, for all x, y U, x R s (y) = y R s (x), trasitive : for all x, y, z U, [xry, yrz] = xrz, for all x, y, z U, [y R s (x), z R s (y)] = z R s (x), for all x, y U, y R s (x) = R s (y) R s (x), Euclidea : for all x, y, z U, [xry, xrz] = yrz, for all x, y, z U, [y R s (x), z R s (x)] = z R s (y), for all x, y U, y R s (x) = R s (x) R s (y). They correspod to the properties of eighborhood operators. For this reaso, we have i fact used the same amig system for both biary relatios ad eighborhood operators. Let R ad Q be two biary relatios. We defie operatios o biary relatios through set-theoretic operatios: R = {(x, y) ot xry}, R Q = {(x, y) xry ad xqy}, R Q = {(x, y) xry or xqy}. (3) They are referred to as the complemet, itersectio, ad uio of biary relatios, respectively. For two relatios R ad Q, the successor eighborhoods defied by R, R Q, ad R Q are give by: ( R) s (x) = R s (x), (R Q) s (x) =R s (x) Q s (x), (R Q) s (x) =R s (x) Q s (x). (4) They follow from the defiitios of operatios o biary relatios ad successor eighborhoods. The set iclusio defies a order o biary relatios o U. A relatio R is said to be fier tha aother relatio Q, i.e., Q is coarser tha R, if R Q. The successor eighborhoods of a fier relatio are smaller tha that of a coarser relatio, amely: 4

5 R Q for all x U, R s (x) Q s (x). (5) Relatio R Q is fier tha both R ad Q, they are fier tha R Q. With a biary relatio, we ca defie additioal types of eighborhoods of x: R p (x) = {y yrx}, R p s (x) = {y xry ad yrx} = R p (x) R s (x), R p s (x) = {y xry or yrx} = R p (x) R s (x). (6) They are called the predecessor, predecessor-ad-successor, ad predecessoror-successor eighborhood operators, respectively. Relatioships betwee these eighborhood system ca be expressed as: R p s (x) R p (x) R p s (x), R p s (x) R s (x) R p s (x). (7) A biary relatio ad eighborhood operators R p ad R s uiquely determie each other, amely, xry x R p (y) y R s (x). (8) However, it is impossible to defie others from operators R p s ad R p s. For a symmetric relatio, all eighborhood operators R p, R s, R p s, ad R p s reduce to the same oe. Relatioships betwee properties of a biary relatio ad the eighborhood operators are summarized i Table 1. The etry i. serial stads for iverse serial, ad the etry serial & i. stads for serial ad iverse serial. For reflexive ad symmetric relatios, all four eighborhood operators have the same property of the correspodig biary relatios. Biary relatio based eighborhoods have bee studied by may authors. Orlowska [9 11,20] viewed R p s (x) as a eighborhood of x. Yao ad Li [24] regarded R s (x) as a eighborhood of x. I geeral, oe may defie additioal eighborhood operators, such as R s, R p, R p s, etc., which are selfexplaatory. The family of all possible eighborhood operators iduced by a biary relatio forms a atomic Boolea algebra, i which the set of atoms cosists of R p s, R p s, R p s, ad R p s. There are i total 16 differet eighborhood operators. Each of them represets a differet iterpretatio of the otio of eighborhoods of a elemet of U. Their additive extesios provide eighborhoods of subsets of U. For a biary relatio R, its iverse R 1 is a biary relatio defied by [2]: yr 1 x xry. (9) 5

6 Table 1 Properties of a biary relatio ad iduced eighborhood operators R R s R p R p s R p s ay symmetric symmetric serial serial i. serial serial & i. i. serial i. serial serial serial & i. reflexive reflexive reflexive reflexive reflexive symmetric symmetric symmetric symmetric symmetric trasitive trasitive trasitive trasitive Euclidea Euclidea Euclidea R is reflexive ad trasitive if ad oly if R 1 is reflexive ad trasitive, respectively. R is a symmetric if ad oly if R = R 1. I this case, we have R = R 1 = R R 1 = R R 1. Idepedet of the properties of R, R R 1 ad R R 1 are symmetric relatios. For a relatio R ad its iverse R 1, the applicatio of operators,, ad produces 16 differet relatios, such as R, R 1, ad R R 1. The predecessor eighborhood defied by R is the successor eighborhood defied by R 1, amely, R p (x) = {y yrx} = {y xr 1 y} = Rs 1 (x). (10) Combiig this result with equatios (4) ad (6), oe ca establish a oeto-oe correspodece betwee differet eighborhood operators iduced by R with the successor eighborhoods iduced by relatios derived from R ad R 1. For example, R p s ad R p s ca be expressed i terms of successor operator as follows: R p s (x) = (R R 1 ) s (x), R p s (x) = (R R 1 ) s (x). (11) We therefore have a alterative method for the formulatio of eighborhood operators with respect to a biary relatio. Neighborhood operators ca be exteded to subsets of uiverse by additive extesio. For example, for the successor eighborhood operator we have: 6

7 R s (X)= R s (x) x X = {y there exists a x X such that xry}, (12) where the same symbol is used to deote both eighborhood operators. The same relatioships betwee eighborhood operators ca be expressed usig subsets of U. For istace, for X U we have: R p (X)=R 1 s (X), R p s (X)=(R R 1 ) s (X), R p s (X)=(R R 1 ) s (X). (13) They are couterparts of equatios (10) ad (11). 3 ROUGH SET APPROXIMATION OPERATORS I this sectio, we review ad geeralize Pawlak approximatio operators based o the otio of 1-eighborhood systems. 3.1 PAWLAK ROUGH SETS Let R U U be a equivalece relatio o U, i.e., R is reflexive, symmetric, ad trasitive. The pair apr = (U, R) is called a Pawlak approximatio space. The equivalece relatio R partitios the uiverse U ito disjoit subsets called equivalece classes. Elemets i the same equivalece class are said to be idistiguishable. Equivalet classes of R are called elemetary sets. A uio of elemetary sets is called a defiable (composed) set [12,13]. The empty set is cosidered to be a defiable set [23]. The family of all defiable sets is deoted by Def(apr). A Pawlak approximatio space defies uiquely a topological space (U, Def(apr)), i which Def(apr) is the family of all ope ad closed sets [12]. Give a subset X U, oe ca approximate X by a pair of subsets of U. The lower approximatio apr(x) is the greatest defiable set cotaied i X, ad the upper approximatio apr(x) is the least defiable set cotaiig X. They correspod to the iterior ad closure of X i the topological space (U, Def(apr)), ad are dual to each other: (a) (b) apr(x) = apr( X), apr(x) = apr( X). 7

8 Oe may iterpret apr, apr : 2 U 2 U as a pair of dual uary set-theoretic operators [6,22]. The system (2 U,,,, apr, apr) is called a Pawlak rough set algebra defied by the equivalece relatio R. It may be viewed as a extesio of classical set algebra (2 U,,, ). I the developmet of rough set theory, two additioal ad distict iterpretatios of approximatio operators have bee proposed. Oe is focused o the elemets of U, ad the other o the equivalece classes of R. A elemet x U belogs to the lower approximatio of X if all its equivalet elemets belog to X. It belogs to the upper approximatio of X if at least oe of its equivalet elemets belogs to X. That is, (i) apr(x)={x U [x] R X} = {x U for all y U, xry implies y X} = {x U y[y [x] R = y X]}, apr(x)={x U [x] R X } = {x U there exists a y U such that xry ad y X} = {x U y[y [x] R, y X]}, where [x] R = {y xry}, (14) is the equivalece class cotaiig x. This iterpretatio of approximatio operators is related to iterpretatio of the ecessity ad possibility operators i modal logic [23,24]. Alteratively, i terms of equivalece classes of R, the pair of lower ad upper approximatio operators ca be defied by: (ii) apr(x) = {[x] R x U, [x] R X}, apr(x) = {[x] R x U, [x] R X }. The lower approximatio of X is the uio of equivalece classes that are subsets of X, ad the upper approximatio is the uio of equivalece classes that have a oempty itersectio with X. If o-equivalece biary relatios are used, the two iterpretatios (i) ad (ii) provide differet geeralizatios of approximatio operators [9 11,16,17,22 25]. They are discussed i the followig subsectios. 8

9 3.2 APPROXIMATION AND NEIGHBORHOOD OPERATORS I geeralizig Pawlak approximatio operators, we may therefore use differet eighborhood operators to defie distict approximatio operators. For a equivalece relatio R, the equivalece class [x] R may be cosidered as a eighborhood of x. Let deote a arbitrary eighborhood operator ad (x) the correspodig eighborhood of x. By substitutig [x] R with (x) i defiitio (i), we defie a pair of approximatio operators [4,6,23]: (I) apr (X)={x U (x) X} = {x U for all y U, y (x) implies y X]} = {x U y[y (x) = y X]}, apr (X)={x U (x) X } = {x U there exists a y U such that y (x) ad y X} = {x U y[y (x), y X]}. The subscript idicates that the approximatio operators are defied based o a particular eighborhood operator. They ca be viewed as a geeralizatio of (i). The system (2 U,,,, apr, apr ) is called a rough set algebra. Theorem 1 For a arbitrary eighborhood operator, the pair of approximatio operators satisfies the properties: (L0) (U0) apr (X) = (apr ( X)), apr (X) = (apr ( X)); (L1) apr (U) = U, (U1) apr ( ) = ; (L2) apr (X Y ) = apr (X) apr (Y ), (U2) apr (X Y ) = apr (X) apr (Y ). PROOF. It ca be easily checked by defiitio that (L1) ad (U1) hold. Approximatio operators are defied by usig the uiversal ad existetial quatifiers. The other two properties trivially follow from the followig laws of predicate logic: (a) (b) x[p(x)] ( x[ P(x)]), x[p(x)] ( x[ P(x)]), x[p(x) Q(x)] x[p(x)] x[q(x)], x[p(x) Q(x)] x[p(x)] x[q(x)]. 9

10 where the variable x is over U, ad P ad Q are predicate symbols. The iterpretatio of rough set approximatio operators usig predicate logic was recetly studied by Wog ad Yao [21]. Properties (L0) ad (U0) show that approximatio operators apr ad apr are dual to each other. Properties with the same umber may be cosidered as dual properties. The first three properties are idepedet. They also imply may other useful properties of approximatio operators, for example, (L3) apr (X Y ) apr (X) apr (Y ), (U3) apr (X Y ) apr (X) apr (Y ); (L4) X Y = apr (X) apr (Y ), (U4) X Y = apr (X) apr (Y ); (L5) apr (X) = apr ( {x}), (U5) x X apr (X) = x X apr ({x}). Properties (L4) ad (U4) state that approximatio operators are mootoic with respect to set iclusio. Properties (L5) ad (U5) show the relatioships betwee the approximatios of the set X ad the approximatios of a family of subsets of U costructed from the sigleto subsets of X. Additioal properties of approximatio operators are determied by the properties of eighborhood operators. The mai results are summarized i the followig theorem. Theorem 2 Suppose : U 2 U is a eighborhood operators. With respect to serial, iverse serial, reflexive, symmetric, trasitive, ad Euclidea eighborhood operators, the approximatio operators have the followig correspodig properties: (L6) apr ( ) =, (U6) apr (U) = U, (LU6) apr (X) apr (X); (L7) for all x U, apr ( {x}) U; (U7) for all x U, apr ({x}) ; (L8) apr (X) X, (U8) (L9) X apr (X); X apr (apr (X)), (U9) apr (apr (X)) X; 10

11 (L10) (U10) (L11) (U11) apr (X) apr (apr (X)), apr (apr (X)) apr (X); apr (X) apr (apr (X)), apr (apr (X)) apr (X)). PROOF. By the duality of approximatio operators, we oly eed to prove oe of the dual properties. Serial eighborhood operator: For ay x U, we have (x). It immediately follows that (L6) holds. Now suppose x apr (X). We have (x) X ad (x). They imply (x) X, amely, x apr (X). Thus, property (LU6) holds. I fact, oe ca easily show that (L6), (U6), ad (LU6) are equivalet, provided that (L0), (U0), (L2), ad (U2) hold. Iverse serial eighborhood operator: For ay x U, there must exist a y U such that x (y). Hece, (y) {x}. By defiitio, y must ot belog to apr ( {x}), which implies that (L7) holds. Reflexive eighborhood operator: For ay x U, x (x). Suppose x apr (X), which is equivalet to (x) X. Combiig x (x) ad (x) X, we have x X. Thus, (L8) holds. Symmetric eighborhood operator: Suppose x X. By the symmetry of, for all y (x) we have x (y), i.e., x (y) X. This implies that for all y (x), y apr (X). Hece, (x) apr (X), which meas that x apr (apr (X)). Therefore, (L9) holds. Trasitive eighborhood operator: Suppose x apr (X), i.e., (x) X. By trasitivity, for all y (x), (y) (x) X. This is equivalet to say that for all y (x), y apr (X). Oe ca therefore coclude that (x) apr (X) ad i tur x apr (apr (X)). That is, (L10) holds. Euclidea eighborhood operator: Suppose x apr (X), i.e., (x) X. By the Euclidea property of, for all y (x), (x) (y). Combiig this result with (x) X, we ca coclude that for all y (x), y apr (X). This is equivalet to say (x) apr (X), which implies x apr (apr (X)). Therefore, (L11) holds. Approximatio operators defied by (I) are cosistet with the otio of ecessity ad possibility operators i modal logic [11,23,24]. With respect to axioms i modal logic [1], if ecessity operator is replaced by apr, possibility operator by apr, egatio by set complemet, cojuctio by set itersectio, disjuctio by set uio, ad implicatio by 11

12 set iclusio, oe ca obtai the axioms of approximatio operators. Most of the properties of approximatio operators discussed so far correspod to axioms i modal logic. With respect to a biary relatio, we may defie distict approximatio operators by usig differet eighborhood operators. The choice of a particular rough set algebra depeds o the applicatio. For eighborhood operators R p (x), R s (x), R p s (x), ad R p s (x), four commoly used pairs of approximatio operators are defied by: (I1) (I2) (I3) (I4) apr Rp (X) = {x U R p (x) X}, apr Rp (X) = {x U R p (x) X }, apr Rs (X) = {x U R s (x) X}, apr Rs (X) = {x U R s (x) X }, apr Rp s (X) = {x U R p s (x) X}, apr Rp s (X) = {x U R p s (x) X }, apr Rp s (X) = {x U R p s (x) X}, apr Rp s (X) = {x U R p s (x) X }. For Pawlak approximatio operators, a equivalece relatio R is used. I this case, four eighborhood operators become the same, i.e., R p (x) = R s (x) = R p s (x) = R p s (x) = [x] R. All defiitios (I1)-(I4) are equivalet. The coditio of a equivalece relatio is sufficiet but ot ecessary. A ecessary ad sufficiet coditio is give below. Theorem 3 Two pairs of lower ad upper approximatio operators from (I1)- (I4) are equivalet if ad oly if the biary relatio R is symmetric. PROOF. ( =) If R is a symmetric relatio, we have for R p (x) = R s (x) = R p s (x) = R p s (x). By defiitio, (I1)-(I4) are equivalet. (= ) Cosider the upper approximatio of sigleto subsets of U. By defiitio, for approximatio operator apr Rp, we have: apr Rp ({x}) = {y R p (y) {x} } Similarly, = {y x R p (y)} = {y xry} = {y y R s (x)} =R s (x). (15) 12

13 apr Rs ({x}) = R p (x), apr Rp s ({x}) = R p s (x), apr Rp s ({x}) = R p s (x). (16) Suppose (I1) ad (I2) are equivalet, we must have: for all x U, R s (x) = apr Rp ({x}) = apr Rs ({x}) = R p (x). (17) Therefore, we ca coclude that R is symmetric. Followig the same argumet, we ca show that the equivalece of ay pair from (I1)-(I4) implies that R is a symmetric relatio. If a biary relatio is ot symmetric, oe may obtai distict approximatio operators. By defiitio, each eighborhood operator defies a pair of dual approximatio operators. Orlowska [11] used defiitio (I1) i the ivestigatio of dyamic iformatio systems ivolvig temporal iformatio. Defiitio (I4) was used by Orlowska [9 11] ad Wasilewska [20]. The operators give by (I2) is a commoly used defiitio, which has bee ivestigated by may authors either explicitly through a biary relatio or implicitly through a coverig of the uiverse [11,16,17,22,24]. A umber of proposals have bee made for usig approximatio operators that are ot dual to each other. Slowiski ad Vaderpoote [19] used apr Rp ad apr Rs as a pair of approximatio operators i the study of rough approximatios usig biary relatios that are oly reflexive. Wybraiec-Skardowska [22] used apr Rs ad apr Rp as a pair of approximatios determied by a biary relatio. I this study, we oly cosider dual approximatio operators. Accordig to property (U2) ad equatios (15) ad (16), upper approximatio operators ca be iterpreted usig eighborhood operators: apr Rp (X) = R s (X), apr Rs (X) = R p (X), apr Rp s (X) = R p s (X), apr Rp s (X) = R p s (X). (18) Clearly, rough set approximatio operators are i fact eighborhood operators. The upper approximatio of X defied by the predecessor eighborhood operator is the successor eighborhood of X, while the upper approximatio of X defied by the successor eighborhood operator is the predecessor eighborhood of X. I cotrast, operators R p s ad R p s produce upper approximatios that are the same as the correspodig eighborhoods. For these operators, we ca establish the followig relatioships: 13

14 apr Rp s (X)={x U apr Rp s ({x}) X}, apr Rp s (X)={x U apr Rp s ({x}) X}. (19) This coectio betwee lower ad upper approximatio operators has bee ivestigated by some authors [17,22]. Wybraiec-Skardowska [22] defied a lower approximatio operator i terms of upper approximatios of sigleto subsets of U usig equatio (19). Oe ca verify that there does ot exist such a relatioship betwee apr Rp ad apr Rp, ad betwee apr Rs ad apr Rs. This implies that i geeral oe may ot use equatio (19) to defie dual approximatio operators. 3.3 APPROXIMATION OPERATORS AND NEIGHBORHOODS I a Pawlak approximatio space, the family of equivalece classes forms a partitio of uiverse. By usig coverigs istead of partitios, may studies geeralize Pawlak approximatio operators [16,22,25]. Such a geeralizatio ca be easily iterpreted i the framework of eighborhood systems. Usig 1-eighborhood systems, we may defie a pair of approximatio operators by replacig the equivalece class [x] R with the eighborhood (x) i defiitio (ii). However, there exists a problem with such a straightforward extesio. The lower ad upper approximatio operators are ot ecessarily dual operators. To resolve this problem, oe may exted defiitio (ii) i two ways. Either the lower or the upper approximatio operator may be exteded, ad the other oe is defied by duality. The results are two pairs of dual approximatio operators: (II ) apr (X)= {(x) x U, (x) X} = {x U y[x (y), (y) X]}, apr (X)= apr ( X) = {x U y[x (y), (y) X]}, = {x U y[x (y), (y) X = ]}, = {x U y[x (y) = (y) X ]}, (II ) apr (X)= apr ( X) = {x U y[x (y), (y) X ]} = {x U y[x (y) = (y) X = ]} = {x U y[x (y) = (y) X]}, apr (X)= {(x) x U, (x) X } = {x U y[x (y), (y) X ]}. 14

15 They were suggested ad studied, together with several pairs of dual approximatio operators, by Pomykala [16,17] usig the otio of coverigs. Properties of these approximatio operators are summarized below. Theorem 4 Suppose : U 2 U is a arbitrary eighborhood operator. Approximatio operators apr ad apr satisfy (L0), (U0), (L8), (U8), (L10), (U10), ad the followig weaker versios of (L2) ad (U2): (L2 ) apr (X Y ) apr (X) apr (Y ), (U2 ) apr (X Y ) apr (X) apr (Y ). Approximatio operators apr ad apr satisfy (L0)-(L2), (U0)-(U2), (L9), (U9), ad the followig more restricted versio of (L8) ad (U8): (L8 ) apr (apr (X)) apr (X), (U8 ) apr (X) apr (apr (X)). PROOF. All these properties ca be easily checked by defiitios (II ) ad (II ). As examples, we prove that apr satisfies (L10), ad apr ad apr satisfy (L9). Assume x apr (X). By defiitio, there must exist a y U such that x (y) ad (y) X. Moreover, for all z (y) we have z apr (X), which implies (y) apr (X). From x (y), we immediately have x apr (apr (X)). Therefore, apr (X) apr (apr (X)), amely, (L10) holds. For a subset X U, assume x X. By defiitio, we have apr (X) = {(z) z U, (z) X }. From assumptio x X, for ay y if x (y) the x (y) X. This implies (y) apr (X). By defiitio, we have x apr (apr (X)), amely (L9) holds. Theorem 5 Suppose : U 2 U is a iverse serial eighborhood operator. Approximatio operators apr ad apr satisfy (L1) ad (U1). Approximatio operators apr ad apr satisfy (L8) ad (U8). PROOF. For a iverse serial eighborhood operator, the family {(x) x U} is a coverig of the uiverse. It easily follows that apr satisfies (L1). For a subset X U, assume x X. Sice is a iverse serial eighborhood operator, there must exist a y U such that x (y). We have x X (y). By defiitio, x apr (X). Thus, apr satisfies (U8). 15

16 Properties (L2 ) ad (U2 ) imply (L3), (L4), (U3), (U4), ad the followig weaker versio of (L5) ad (U5): (L5 ) apr (X) x X (U5 ) apr (X) x X apr ( {x}), apr ({x}). Properties (L8) ad (U8) imply (LU6), (L7), (U7), ad (L8 ) ad (U8 ). Properties (L2) ad (U2) imply (L3)-(L5), (U3)-(U5), ad (L2 ) ad (U2 ). By Theorems 4 ad 5, the correspodig approximatio operators satisfy such implied properties. For a arbitrary eighborhood operator, the family of eighborhoods {(x) x U} may ot be a coverig of the uiverse. It is a coverig of U if ad oly if is a iverse serial eighborhood operator. Our formulatio is therefore more geeral tha the existig defiitios usig the otio of coverigs. Accordig to Table 1, for a arbitrary biary relatio R, oe of the four families, C Rp = {R p (x) x U}, C Rs = {R s (x) x U}, C Rp s = {R p s (x) x U}, C Rp s = {R p s (x) x U}, (20) ecessarily forms a coverig of U. The families C Rs ad C Rp s are coverigs if R is a iverse serial. If the relatio R is serial, C Rp ad C Rp s are coverigs. If R is reflexive, all four families are coverigs. Theorem 5 provides additioal properties satisfied by approximatio operators costructed from a coverig. With respect to a coverig of the uiverse, Zakowski [25] used apr ad apr as a pair of approximatio operators. The same otio was also adopted by Wybraiec-Skardowska [22]. Orlowska [9 11] used the pair of of approximatio operators apr R p s ad apr R p s. A problem with such defiitios is that they may ot produce dual operators [3,16]. 4 CONNECTIONS OF APPROXIMATION OPERATORS With respect to a eighborhood operator, we have defied three pairs of dual approximatio operators by usig (I), (II ), ad (II ). For compariso, we restate them usig a similar format as follows: 16

17 Table 2 A example of approximatio operators X apr apr apr apr apr apr {a} {a} {a} {a, b} {b} {c} {a,c} {b} {a,b} {a,b} {c} {b} {b} {c} {c} {c} {c} {a,b} {a,c} {a,c} {a,b} {a,b} {a,b} {a,b} {a,c} {b} {a,b} {c} {a,c} {c} U {b, c} {b, c} U {b, c} U {c} U U U U U U U U (I) apr (X)={x U (x) X}, apr (X)={x U (x) X }; (II ) apr (X)={x U y[x (y), (y) X]}, apr (X)={x U y[x (y) = (y) X ]}; (II ) apr (X)={x U y[x (y) = (y) X]}, apr (X)={x U y[x (y), (y) X ]}. Defiitio (I) oly uses the eighborhood of x to decide if x belogs to the lower or the upper approximatio. I makig the same decisio, both defiitios (II ) ad (II ) use a family of eighborhoods {(y) y U, x (y)}. I some sese, they may be regarded as coverse defiitios. While defiitio (II ) uses the existetial quatifier for lower approximatio ad the uiversal quatifier for upper approximatio, defiitio (II ) uses the opposite quatifiers. These three defiitios produce distict but related approximatio operators. Example 6 Cosider a uiverse U = {a, b, c}. Suppose a eighborhood operator is give by: (a) = {a, b}, (b) = {c}, (c) = {b}. It is a serial ad a iverse serial eighborhood operator. Usig defiitios (I), (II ), ad (II ), we have three pairs of approximatio operators i Table 2. 17

18 From the example, oe may observe that three pairs of approximatio operators are differet, although they are defied based o the same eighborhood operator. Operators apr ad apr give tighter approximatios tha that of apr ad apr. For other operators, there is ot such a relatioship. If additioal properties are imposed o eighborhood operators, much stroger relatioships may be established. Theorem 7 Suppose : U 2 U is a eighborhood operator. If is iverse serial, the followig relatioships hold: (R1) apr (X) apr (X) X apr (X) apr (X). If is a reflexive, we have: (R2) apr (X) apr (X) apr (X) X apr (X) apr (X) apr (X). PROOF. (R1): For a iverse serial eighborhood operator, by Theorems 4 ad 5 we kow that the set X lies betwee its lower ad upper approximatios as defied by (II ) ad (II ). We oly eed to show apr (X) apr (X). The other relatio apr (X) apr (X) ca be obtaied by duality. Assume x apr (X). Sice is iverse serial, there must exist a y U such that x (y). By assumptio x apr (X) ad defiitio of apr, we ca coclude that (y) X. It follows that x apr (X). Therefore, apr (X) apr (X). (R2) ca be similarly proved. Theorem 8 [16, page 660, Corollary of Theorem 3] Two pairs of lower ad upper approximatio operators defied by (II ) ad (II ) are equivalet if ad oly if the family {(x) x U} forms a partitio of the uiverse. Theorem 9 Two pairs of lower ad upper approximatio operators defied by (I) ad (II ) are equivalet if ad oly if the eighborhood operator is reflexive ad trasitive. PROOF. ( =) By the reflexivity of, we have apr (X) apr (X). Now we eed to prove the reverse. Assume x apr (X). There must exist a y U such that x (y) ad (y) X. By the trasitivity of, we have (x) (y) X, which implies x apr (X). Therefore, apr (X) = apr (X). (= ) Suppose apr (X) = apr (X) ad apr (X) = apr (X) for all X U. For ay x U, by defiitio we have x apr ({x}). By assumptio, 18

19 x apr ({x}). It follows x (x), that is, is reflexive. For three elemets x, y, z U, assume y (x) ad z (y). By defiitio, y apr ((x)). Thus, y apr ((x)). This implies (y) (x). Combiig with z (y), we have z (x). Therefore, is trasitive. Theorem 10 Two pairs of lower ad upper approximatio operators defied by (I) ad (II ) are equivalet if ad oly if the eighborhood operator is symmetric ad trasitive. PROOF. ( =) Assume x apr (X). Cosider the family O(x) = {y U x (y)}. If O(x) =, by the symmetry of we ca coclude (x) =, hece x apr (X). If O(x), by defiitio of apr we have x (y) ad (y) X for all y O(x). By the trasitivity of, we have (x) (y) X. Thus, x apr (X). By summarizig the results for both cases, we obtai apr (X) apr (X). Assume x apr (X). We have (x) X. Cosider ay y U with x (y). By the symmetry of, it follows y (x). By the trasitivity of, we ca coclude that (y) (x) X. Accordig to defiitio, x apr (X). We thus proved apr (X) apr (X). Therefore, apr (X) = apr (X). (= ) Suppose apr (X) = apr (X) ad apr (X) = apr (X) for all X U. For a pair x, y U, assume x (y). By Theorem 4, apr ad apr satisfy (L9). It follows that y apr (apr ({y})). Thus, y apr (apr ({y})). From defiitio, we have (y) apr ({y}). By assumptio, x (y) apr ({y}). It implies y (x), amely, is symmetric. For three elemets x, y, z U, assume y (x), z (y). By defiitio, x apr ((x)). Hece x apr ((x)). By defiitio, for all w, x (w) implies (w) (x). From the symmetry of ad assumptio y (x), we have x (y), ad hece (y) (x). Combiig this with z (y), we obtai z (x). Therefore, must be trasitive. Based o the results of the last three theorems, Figure 1 summarizes the coditios uder which two pairs of approximatio operators are equivalet. With the coditio: (P) {(x) x U} is a partitio, we have: [reflexivity, P] = symmetry, [symmetry, trasitivity, P] = reflexivity, [reflexivity, symmetry, trasitivity] = P. (21) 19

20 apr, apr Reflexive Trasitive Symmetric Trasitive apr, apr apr, apr {(x) x U} is a partitio Fig. 1. Relatioships betwee approximatio operators It should be poited out that oe of reflexivity, symmetry, ad trasitivity is ecessary for (P). For example, cosider a uiverse U = {a, b, c} with a eighborhood operator give by: (a) = {b}, (b) = {c}, (c) = {a}. Obviously, is ot reflexive, or symmetric ad trasitive. Nevertheless, the family {{b}, {c}, {a}} is a partitio of U. A set of ecessary ad sufficiet coditio for (P) is: (P1) is iverse serial, (P2) for all x, y U, either (x) = (y) or (x) (y) =. Coditio (P1) is equivalet to sayig that x U (x) = U. The followig Corollary follows from the last three theorems. Corollary 11 Three pairs of lower ad upper approximatio operators defied by (I), (II ), ad (II ) are equivalet if ad oly if is reflexive, symmetric, ad trasitive. Neighborhood operators may be iterpreted by usig biary relatios. Their properties are determied by the properties of biary relatios. The results developed i this sectio may be alteratively stated usig biary relatios. 20

21 5 CONCLUSION We propose a biary relatio based framework for the study of eighborhood systems ad rough set approximatios. Withi this framework, these two otios may be formulated, iterpreted, ad compared. The class of 1- eighborhood systems, i which each elemet has oe eighborhood, are studied i detail. A biary relatio is used to defie ad iterpret such eighborhood systems. I particular, 16 distict eighborhood operators are obtaied from a biary relatio. Properties of eighborhood operators are related to the properties of biary relatios. Three geeralizatios of Pawlak approximatio operators are suggested. Each of them captures differet aspects i approximatig a subset of the uiverse. Oe geeralizatio is related to the otio of modal operators i modal logic. For this geeralizatio, oly the eighborhood of x is used to decide the memberships of x i the lower ad upper approximatio of a subset X. For the other two geeralizatios, the family of eighborhoods, {(y) x (y)}, is used to decide the memberships of x. Coditios o eighborhood operators are idetified, uder which some or all of these approximatio operators are equivalet. The otios of eighborhood systems ad approximatio operators are useful i modelig approximatio. I this paper, we oly aalyzed 1-eighborhood systems. A careful study of geeral eighborhood systems may produce a more powerful ad useful tool i approximatios. ACKNOWLEDGEMENTS The author is grateful for discussio with ad suggestios from T.Y. Li o eighborhood systems. Refereces [1] B.F. Chellas, Modal Logic: A Itroductio, Cambridge Uiversity Press, Cambridge, [2] P.M. Coh, Uiversal Algebra, Harper ad Row Publishers, New York, [3] L.T. Koczy, Review of O a cocept of rough sets by Zakowski, Mathematical Review, 84i:03092 (1984). [4] T.Y. Li, Topological ad fuzzy rough sets, i: R. Slowiski (Ed.), Itelliget Decisio Support: Hadbook of Applicatios ad Advaces of the Rough Sets Theory, Kluwer Academic Publishers, Bosto, 1992, pp

22 [5] T.Y. Li, Neighborhood systems applicatio to qualitative fuzzy ad rough sets, i: P.P. Wag (Ed.), Advaces i Machie Itelligece & Soft- Computig, Departmet of Electrical Egieerig, Duke Uiversity, Durham, North Carolia, USA, 1997, pp [6] T.Y. Li ad Q. Liu, Rough approximate operators: axiomatic rough set theory, i: W.P. Ziarko (Ed.), Rough Sets, Fuzzy Sets ad Kowledge Discovery, Spriger-Verlag, Lodo, 1994, pp [7] T.Y. Li, Q. Liu, K.J. Huag, ad W. Che, Rough sets, eighborhood systems ad approximatio, i: Z.W. Ras, M. Zemakova, ad M.L. Emrichm (Eds.), Methodologies for Itelliget Systems, 5: Proceedigs of the fifth Iteratioal Symposium o Methodologies of Itelliget Systems, Koxville, Teessee, October 25-27, 1990, North-Hollad, New York, pp [8] T.Y. Li ad Y.Y. Yao, Miig soft rules usig rough sets ad eighborhoods, i: Proceedigs of the Symposium o Modellig, Aalysis ad Simulatio, Computatioal Egieerig i Systems Applicatios (CESA 96), IMASCS Multicoferece, Lille, Frace, July 9-12, 1996, pp [9] E. Orlowska, Sematics of odetermiistic possible worlds, Bulleti of the Polish Academy of Scieces: Mathematics, 33: (1985). [10] E. Orlowska, Sematics aalysis of iductive reasoig, Theoretical Computer Sciece, 43:81-89 (1986). [11] E. Orlowska, Kripke sematics for kowledge represetatio logics, Studia Logica, 49: (1990). [12] Z. Pawlak, Rough sets, Iteratioal Joural of Computer ad Iformatio Sciece, 11: (1982). [13] Z. Pawlak, Rough classificatio, Iteratioal Joural of Ma-Machie Studies, 20: (1984). [14] Z. Pawlak, Rough sets: a ew approach to vagueess, i: L.A. Zadeh ad J. Kacprzyk (Eds.), Fuzzy Logic for the Maagemet of Ucertaity, Joh Wiley & Sos, New York, 1992, pp [15] Z. Pawlak, Hard ad soft sets, i: W.P. Ziarko (Ed.), Rough Sets, Fuzzy Sets ad Kowledge Discovery, Spriger-Verlag, Lodo, 1994, pp [16] J.A. Pomykala, Approximatio operatios i approximatio space, Bulleti of the Polish Academy of Scieces, Mathematics, 35: (1987). [17] J.A. Pomykala, O defiability i the odetermiistic iformatio system, Bulleti of the Polish Academy of Scieces, Mathematics, 36: (1988). [18] W. Sierpeski ad C. Krieger, Geeral Topology, Uiversity of Toroto, Toroto,

23 [19] R. Slowiski ad D. Vaderpoote, Similarity relatio as a basis for rough approximatios, i: P.P. Wag (Ed.), Advaces i Machie Itelligece & Soft- Computig, Departmet of Electrical Egieerig, Duke Uiversity, Durham, North Carolia, USA, 1997, pp [20] A. Wasilewska, Coditioal kowledge represetatio systems model for a implemetatio, Bulleti of the Polish Academy of Scieces: Mathematics, 37:63-69 (1987). [21] S.K.M. Wog ad Y.Y. Yao, Roughess theory, i: Proceedigs of the Sixth Iteratioal Coferece o Iformatio Processig ad Maagemet of Ucertaity i Kowledge-Based Systems, Graada, Spai, July 1-5, 1996, pp [22] U. Wybraiec-Skardowska, O a geeralizatio of approximatio space, Bulleti of the Polish Academy of Scieces: Mathematics, 37:51-61 (1989). [23] Y.Y. Yao, Two views of the theory of rough sets i fiite uiverses, Iteratioal Joural of Approximate Reasoig, 15: (1996). [24] Y.Y. Yao ad T.Y. Li, Geeralizatio of rough sets usig modal logic, Itelliget Automatio ad Soft Computig, a Iteratioal Joural, 2: (1996). [25] W. Zakowski, Approximatios i the space (U, Π), Demostratio Mathematica, XVI: (1983). 23