On General Binary Relation Based Rough Set

Transcription

1 SSN , England, K Journal of nformation and Computing Science Vol. 7, No. 1, 2012, pp On General Binary Relation Baed Rough Set Weihua Xu 1,+, Xiantao Zhang 1, Qiaorong Wang 1 and Shuai Sun 2 1 School of Mathematic of Statitic, Chongqing niverity of Technology, Chongqing, P.R. China 2 Bureau of Statitic, Lijin, Dongying, P.R. China (Received September 15, 2011, accepted September 18, 2011) btract. To keep the key idea of rough et and the repreentation of information in rough et theory, empty repreentation i proceed properly and three new form of rough approximation et are defined a a generation of general binary relation baed rough et model. Moreover, the propertie of approximation operator in thee new rough et are dicued. The relation among them are tudied in thi paper. n addition, example are arranged to interpret what we tudied in thi paper. Keyword: General binary relation, pproximation operator, Rough et, Empty decription, Duality 1. ntroduction Rough et theory wa firt propoed by Pawlak Z in t i a mathematical tool to proce information with uncertainty and vaguene. nd it i alo a ueful oft computing tool in intelligence computing. The rough et theory can yet be regarded a a kind of more effective method to handle complex ytem in data mining (DM) and knowledge dicovery in databae (KDD) [1,3,4,5]. Rough et theory, probability theory, fuzzy et theory and evidence theory are all tool to deal with uncertainty. Compared with other theorie, the mot ignificant difference i that no more prior information i needed but the pecified information ytem for problem in rough et theory. Much better affection may come about in practical problem by combining rough et and other method. Noie i a factor which can t be avoided in practice. nfluenced by noie in data and limited by the requet of practical problem, the original rough et propoed by Pawlak i confined in practical application. Many generalized rough et model have been propoed and tudied ytematically. The popularization uch a variable preciion rough et, dominance-baed rough et, tolerance-baed rough et, fuzzy rough et, rough et baed on covering, rough et baed on evidence theory, probabilitic rough et, etc.[1,2,8,9,10,11,12,14], make that rough et theory affect in more area and field. More ueful information i being dicovered and more value are being produced in real world. reearche on rough et are expanding in depth and further, the rough et theory i now being more and more abundant, theorized and ytematic. Succeful application have been applied in many area and field, uch a in ubject medical cience, chemitry, material cience, geographical cience, management, finance, conflict reolution, and o on. Excellent effect have been ucceeded in many area. Requirement in application and the propelling of achievement are promoting rough et theory to be one of the mot active reearch area in information cience and everal interdicipline [12,14,15]. Studying on general binary relation baed rough et can make rough et theory more adaptable for generalized relation and produce more value in practice. Theorie on general binary relation baed rough et, which i denoted by GBRS, have been placed to ome extent [6,7,14]. On the bai of Pawlak rough et, equivalence relation are popularized to general binary relation. nd generalization of GBRS are tudied by contructive method, axiomatic method and the key idea of rough approximating in thi paper. Propertie of the correponding approximation operator are dicued and proved. Moreover, example are employed to help undertand what we tudy in thi paper. 2. Pawlak Rough Set The claical rough et propoed by Pawlak i on the bai of equivalence relation on univere [3,4,5]. + Correponding author. Tel.: addre: chxwh@gmail.com. Publihed by World cademic Pre, World cademic nion

2 Journal of nformation and Computing Science, Vol. 7 (2012) No. 1, pp Object carry ome unique information under relation and they can be claified by equivalence relation. Object claified a the ame cla carry the information which i totally the ame. f a concept can be preented by the union of ome clae, that i the concept equal to the combination of ome clae, then the concept carry all the information repreented by the object in thee clae. Ele, if a concept can t equal to the union of ome clae, a pair of et, which are contructed by the union of ome clae, are employed to repreent the concept approximately. One of the two et i conited of object with totally confirm information that the concept carrie. The election of object in thi form relie on the preciely incluion of clae to the concept. The other et i conited of object with information that poibly upport the concept. The election of thee object conider the nonempty joint between clae and the concept. The pair of et i called, repectively, lower approximation et and upper approximation et [3,4,5,9,11,12,13]. Some baic definition in Pawlak rough et will be illutrated for ue in thi paper. Definition 2.1([3,14]) Let be a nonempty finite et conited of object and called univere. For any x, x i called an object. R i a binary relation on univere. x ha the relation R with y if and only if ( x, y) R, that i xry ( x, y) R. f R atifie (1) Reflective: x, xrx; (2) Symmetric: xry yrx ; (3) Tranitive: xry, yrz xrz ; then R i an equivalence relation on the univere. Definition 2.2([3,14]) Let be the univere. R i an equivalence relation on the univere. Then (, R ) i called Pawlak approximation pace. For any X, X i called a concept on the univere. For any x, [ x] R= { y ( xy, ) R} i called the equivalence cla of x with repect to R. R / = {[ x] R x } i called the partition induced by R to. Definition 2.3([4,14]) Let be the univere. R i an equivalence relation on the univere. For any X, the lower approximation and upper approximation of X with repect to Pawlak approximation pace (, R ) are defined, repectively, a RX = { xx [ ] RX}, RX = { xx [ ] X }. X i definable with repect to R if and only if RX = RX. Ele, X i rough with repect to R if and only if RX RX. R and R are called, repectively, the lower approximation operator and upper approximation operator with repect to R. Thi model i Pawlak rough et model. f [ x] R X, we uually ay that x i preciely upporting the concept X with repect to R. Correpondingly, if [ x] R X, it i aid that x i poibly upporting the concept X with repect to R. Theorem 2.1([3,14,15]) Let be the univere. R i an equivalence relation on the univere. For any X, Y, the following propertie of lower and upper approximation operator hold. (1) RX XRX ; (2a) R(~ X) = ~ R( X); (2b) R(~ X) = ~ R( X); (3a) R( = ) R( = ) ; (3b) R = R = ; (4a) XY R( X) R( Y); (4b) XY R( X) R( Y); (5a) RX ( Y) = RX RY ; (5b) RX ( Y) = RX RY ; (6a) RX ( Y) RX RY ; (6b) RX ( Y) RX RY. Clae are the ueful decription to characterize concept in rough et theory. Poible decription R JC for ubcription: publihing@w.org.uk

3 56 Weihua Xu, et al: On General Binary Relation Baed Rough Set with more uncertainty information hould be not le than compatible decription with preciely upport. Thu, the key idea to repreent concept by a pair of approximation et hould atify RX RX for any X in rough et theory. an important property in rough et theory, the duality hould alo be conidered in the generalized model. Propertie on generalized approximation operator hould take thoe in the above theorem a reference and model. ncertainty in rough et theory i due to the exitence of boundary region. Boundary i a et conited of object with decription upport neither X nor X with repect to the approximation pace. The uncertainty of concept can be reflected and meaured by roughne and accuracy. nd the detailed form of thee uncertainty meaure won t be arranged in thi paper. More about rough et theory can be referred in reference [3,4,5,12,13,14,15]. Reader who need can look back into relative reference and tudy in further. 3. Contruction of General Binary Relation Baed Rough Set Since the equivalence relation limit the application of Pawlak rough et, generalized rough et model have been propoed and tudied. Equivalence relation can be replaced by general binary relation and general binary relation baed rough et ha been contructed. The general binary relation baed rough et (denoted hortly by GBRS) which wa propoed by the contructive method will be introduced in thi ection. Definition 3.1([14,15]) Let be the univere. R i a general binary relation on the univere. For any x, denote R = { y xry }, then R ( x ) i called the ucceor neighborhood of x with repect to R. ny y R i called the ucceor of x with repect to R. For any y, denote Rp ( y) = { x xry }, then Rp ( y ) i called the proceor neighborhood of y with repect to R. ny x Rp ( y) i called the proceor of y with repect to R. From the above definition, the ucceor neighborhood R ( x ) i conited of all object y which atifie xry with x and the proceor neighborhood Rp ( y ) i the et of all object x atifying xry with y. They can both be treated a clae induced by the general binary relation R. ccording to the contructive method of rough et theory, approximation operator in the following form can be defined. Definition 3.2([14,15]) Let be the univere. R i a general binary relation on the univere. Denote = (, R) and it i call a generalized approximation pace. For any X, the lower approximation et and upper approximation et of X with repect to can be defined a follow. () ( X ) = { x R X}, X i definable with repect to if and only if only if ( X ) ( X ). ( X ) = { x R X }. ( X ) = ( X ). Ele, X i rough with repect to if and For convenience to tudy in thi paper, GBRS howed in thi form i called the firt type of GBRS and denoted by GBRS(). The approximation et in GBRS() i defined by ucceor neighborhood and they can alo be defined by proceor neighborhood imilarly. We jut need to replace R ( x ) by Rp ( x ) in the above definition. The approximation defined in thee two form are totally imilar. Definition and propertie of approximation operator are not affected in dicuion. The form of GBRS() defined by proceor neighborhood will be not intend to be dicued in thi paper. Studie on GBRS() can be reviewed in reference [14,15]. We jut preent the propertie of approximation operator and more detail can be looked back into thee reference to tudy further. Theorem 3.1([14,15]) Let be the univere. R i a general binary relation on the univere. For any X, Y, the following propertie of lower and upper approximation operator hold in GBRS(). JC for contribution: editor@jic.org.uk

4 Journal of nformation and Computing Science, Vol. 7 (2012) No. 1, pp a ( X ) ( X ); 1b ( X ) ( X ); 2a ( ) ( ) ; 2b ; 3a X Y ( X ) ( Y ); 3b X Y ( X) ( Y); 4a ( X Y ) ( X ) ( Y ); 4b ( X Y) ( X ) ( Y ); 5a ( X Y ) ( X ) ( Y ); = = = = = = 5b ( X Y ) ( X ) ( Y ). From the decription, which are the repreentation form of information and ued in form of clae in rough et theory, while R ( x ) =, x i conidered carrying empty information and the ucceor neighborhood of x with repect to R i an empty decription. Empty information i alo an information form and empty decription may need to be conidered in concept characterizing. Empty decription are conidered to upport any concept preciely in the above Definition 3.1. That i, if R =, x i regard a an object which upport any concept preciely with repect to R ince R = X hold for any X. Thu, any x uch that R = i included in the lower approximation et of arbitrary concept. But in any upper approximation et, all x atify R = are not included for x appr ( X ) R ( x ) X. That i, for any x appr ( X ), R. The empty decription are conidered in the lower approximation but they are left out in upper approximation. Hence, we have that the property ( X ) ( X ) doe not hold for an arbitrary general binary relation in GBRS(). To undertand GBRS() in the key idea of Pawlak rough approximation, object preciely upport any concept don't have the capability to poibly upport all concept in GBRS(). Thi concluion i illogical and unreaonable in application. However, GBRS() i till a very ueful rough et model for being contruct directly from Pawlak rough et. t trong uit i that the approximation operator till atify duality. So GBRS() can poe better generalization ability for binary relation atify particular propertie uch a reflexive and erial. Example may be vivid to expre the propertie analyzed above and the following example i employed to interpret GBRS() practicality. Example 3.1 Let = { x1, x2, L, x5} be the univere. R i a general binary relation on the univere. R={( x1, x2),( x1, x4),( x2, x3),( x2, x5),( x3, x3),( x3, x4),( x5, x4),( x5, x 5)}. = (, R) i the generalized approximation pace with repect to R on the univere. X1= { x1, x2, x4}, X2= { x3, x4, x5}. Calculate the lower and upper approximation of X1, X 2 uing GBRS(). ccording to Definition 3.1, we have that R ( x1) = { x2, x4}; R ( x2 ) = { x3, x5}; R ( x3 ) = { x3, x4}; R ( x4 ) = ; R ( x ) = { x, x }. 5 From Definition 3.2, the lower and upper approximation of X 1, X 2 can be obtained and lited in the following. 4 5 JC for ubcription: publihing@w.org.uk

5 58 Weihua Xu, et al: On General Binary Relation Baed Rough Set ( X ) = { x, x }; ( X1) = { x1, x3, x5}; ( X ) = { x, x, x, x }; ( X ) = { x, x, x, x, x } Then we have that ( X ) ( X ) don t hold in GBRS(). Moreover, propertie can be verified from thi example and we don t illutrate them in detail. 4. GBRS with Empty Decription Left Out ccording to the analyi of GBRS(), we generalize general binary relation baed rough et and dicu the propertie of the new approximation operator in the following ection. From Section 3, the empty information or empty decription can be dipoed uch that the approximation operator adapt the rough et theory logically. Then, general binary relation baed rough et can be generalized by empty decription being regarded a upporting none concept in rough et theory. The empty decription hould be left out via thi treatment to the empty information. Therefore, the lower and upper approximation et are both conited of object bear nonempty information with repect to general binary relation. That i, x ( X ) R and x X R x hold. Whereupon, we have the following new definition of general binary relation baed rough et with repect to the generalized approximation pace = (, R). Definition 4.1 Let be the univere. R i a general binary relation on the univere. For any X, the lower and upper approximation of X with repect to the generalized approximation pace are defined, repectively, a () ( X ) = { x ( R X ) ( R )}, X i definable with repect to if and only if ( X ) = { x R X }. ( X ) = ( X ). Ele, X i rough with repect to if and only if ( X ) ( X ). Thi form of rough et i called the econd type of general binary relation baed rough et and denoted by GBRS(). Correponding to tudie in the front ection, the propertie of approximation operator in GBRS() can be dicued and tudied further in the following theorem. Theorem 4.1 Let be the univere. R i a general binary relation on the univere. For any X, Y, the following propertie of lower approximation operator and upper approximation operator hold with repect to generalized approximation pace. (1a) ( X ) ( X ); (1b) R X R ; Proof. (1) While x ( X ) x ( X ) = = = X Y X Y X Y X Y X Y = = (2a) ; (2b) ; (3a) ; (3b) ; (4a) ( X ) ( Y ); (4b) ( X Y ) ( X ) ( Y ); (5a) ( X Y ) ( X ) ( Y ); (5b) ( X Y ) ( X ) ( Y ). ( X ) =, thi item i obviou. ume that ( X ), then we have JC for contribution: editor@jic.org.uk

6 Journal of nformation and Computing Science, Vol. 7 (2012) No. 1, pp x( X) ( R X) ( R ) R X x ( X ). Thi item ( X ) ( X ) i proved. tem (1b) can be proved directly from Definition 4.1. (2a) x, R R = and R =. nd then ( = ) { x ( R ) ( R )} = { x ( R = ) ( R )} =, ( = ) { x R = }. (2b) x, R R and R. Then we have ( ) = { x ( R ) ( R )} = { x R }, ( ) = { x R } = { x R }. Hence, ( ) = ( ) i proved. Moreover, from { x R } { x R = } =,we can eaily obtain that ( ) = ( ). (3a) X Y, while Thi item i proved. (3b) X Y, while ( X ) =, the item i obviou. Suppoe that x ( X ) R ( X ) X R ( X ) Y x ( Y ). ( X ) =, thi item i apparent. For ( X ), then we have ( X ), we have that x ( X ), R( X ) X R( X ) Y x ( Y ). Thi item i proved. (4a) From property (3a) in thi theorem, one can eaily have that ( X Y ) ( X ) ( Y ). While ( X ) ( Y ) =, thi property hold obviouly. ume that x( X) ( Y) ( x ( X)) ( x ( Y)) [( R X) ( R )] ( [ R Y) ( R )] ( R X) ( R ) ( R Y) ( R XY) ( R ) x ( X Y ). ( X ) ( Y ), then we have The property ( X Y ) ( X ) ( Y ) i obtained. Hence, thi item i proved. nd item (4b) can be proved imilarly a item (4a) in thi theorem. Propertie (5a) (5b) can be proved directly from item (3a)(3b) in thi theorem. By the comparion of the propertie above with thoe in Pawlak rough et, one can have that the duality doen't hold in GBRS(). The item (1) in Theorem 2.1 correpond to item (1a) (1b) in Theorem 4.1 ince Pawlak rough et can be defined by two equivalence form. Conidering the ignification of Pawlak rough et, the approximation to concept goe along in term of the relation R. Object in the approximation et employ the information and decription they are bearing with repect to the relation R to characterize the approximated concept. n object itelf belong to a concept or not relie on if it poee the information, which preciely upport the concept with repect to the relation R. That i, R X x appr ( X ) hold but doen't hold in GBRS(). So, there exit no affirmative incluion between the y R x X y appr X lower approximation et and the approximated concept in general binary relation baed rough et. Similarly, JC for ubcription: publihing@w.org.uk

7 60 Weihua Xu, et al: On General Binary Relation Baed Rough Set there i no affirmative incluion between the upper approximation et and the approximated concept in GBRS. We can only have that the property lower approximation et i included in upper approximation et hold in GBRS(). nd thi comport a item (1a) in Theorem 4.1. Object in clae undertake the information which are ued to depict concept directly in problem conulting with rough et. mong thee vector, one preciely upport a concept can't be more than thoe in the depicted concept itelf with repect to the relation R. Thi comport a item (1b) in Theorem 4.1. Baed on thee analyi, GBRS() ha the capability to repreent concept approximately by mean of ignoring empty decription. t can make the approximation to concept feaible in ene of logic and practice. t the ame time, a fly in the ointment i that the approximation operator in GBRS() diatify the duality. pproximation operator which meet duality tranform ynchronouly. The tranformation i antithetical and obey duality. While the lower approximation decreae, the upper approximation increae and act in accordance with the lower approximation' tranformation. While the lower approximation increae, the upper approximation decreae correpondingly to the tranform of the lower approximation. Contrarily, the dual ynchronou change of lower approximation hold while the upper approximation tranform. Regrettably, the dual ynchronou change of lower and upper approximation can't hold while concept are depicted with repect to arbitrary general binary relation in GBRS(). However, GBRS() i ueful theoretically and can affect much in application. We till employ Example 3.1 to explain GBRS() in thi ection and the reult are preented in the following. Example 4.1 (Continued from Example 3.1) From Example 3.1, calculate the lower and upper approximation and clear that the duality doen t hold in GBRS(). ccording Definition 4.1 and Example 3.1, the approximation are lited a follow. ( X ) = { x }; nd X1 x3 x5 X2 x1 x2 1 1 ( X1) = { x1, x3, x5}; ( X ) = { x, x, x }; ( X ) = { x, x, x, x, x } = {, }, = {, }. Furthermore, we have that ( X ) = { x }; 1 2 ( X1) = { x2, x3, x5}; ( X ) = ; ( X ) = { x } Then, we can ee that the duality doen t hold in GBRS(). Other propertie in Theorem 4.1 can be verified and they are not arranged in thi paper. 5. GBRS with Empty Decription Taken into ccount Taking GBRS() a a foundation, if object with empty decription are all conidered to upport the any concept preciely with repect to the relation R, then they hould poibly upport the concept with repect to R according to the meaning and key idea of Pawlak rough approximating. Therefore, GBRS() can be generalized with empty decription taken into account in both lower and upper approximation. Then GBRS in the following form can be defined and dicued. Definition 5.1 Let be the univere. R i a general binary relation on the univere. For any X, the lower and upper approximation of X with repect to generalized approximation pace are defined, repectively, a () ( X ) = { x R X}, ( X ) = { x ( R X ) ( R = )}. X i definable with repect to if and only if ( X ) = ( X ). Ele, X i rough with repect to if and only if ( X ) ( X ). Thi form of rough et i called the third type of general binary relation baed JC for contribution: editor@jic.org.uk

8 Journal of nformation and Computing Science, Vol. 7 (2012) No. 1, pp rough et and denoted by GBRS(). Similarly a the above dicuion, propertie of approximation operator in GBRS() can be tudied and preented in the following theorem. Theorem 5.1 Let be the univere. R i a general binary relation on the univere. For any X, Y, the following propertie of lower and upper approximation operator with repect to generalized approximation pace hold in GBRS(). (1a) ( X ) ( X ); (2b) ( ) = ( ) = ; (1b) R X R ; (3a) X Y ( X ) ( Y ); x ( X) x ( X ) x R x = X x X m xr x = Xk Xk x k= 1 m m Xk= x R x = = Xk k= 1 k= 1 = (1c),, ; (1d) { }, ; (1e) { }, ( X ); (2a) ; Proof. (1a) For any x ( X ), while R x =, we have that Hence, ( X ) ( X ) i proved. k = = (3b) X Y ( X ) ( Y ); (4a) ( X Y ) ( X ) ( Y ); (4b) ( X Y ) ( X ) ( Y ); (5a) ( X Y ) ( X ) ( Y ); (5b) ( X Y) ( X ) ( Y). x i obviou and thi item hold. ume that R x ( X ) R X R X x ( X ). tem (1b) (1c) can be proved directly from Definition 5.1. (1d) For any x { xr = }, R ( x ) = hold. Then, one can have that x m k k k= 1 k k R ( x ) = ( X ),( X ) x ( X ),( X ). Hence, thi property i proved. (2a) ccording to Definition 5.1, we can eaily obtain that ( = ) { xr = } { xr = }, ( = ) { x ( R ( x ) ) ( R ( x ) = )} = { xr = }, x, then Thu, we have ( = ) ( ) hold. f there exit any x uch that R x =, then we have that (. ) Ele ( )=. Hence, thi item ( ) = i proved. (2b) For any x, R hold. Furthermore, the following procee hold. ( ) = { x R } =, ( ) = { x ( R ) ( R = )} = { xr }, Then, thi item i proved. (3a) Since X Y, for any x ( X ), we have that R x X Y. Obviouly, x ( Y ) hold and thi item i proved. (3b) For any x ( X ), while R ( x ) =, x ( Y ) i obviou. Suppoe R, then we have R X. X Y ha been known, one can acquire that R Y. That i, Thi item i proved. (4a) We can eaily have that x ( Y ) hold. Hence, JC for ubcription: publihing@w.org.uk

9 62 Weihua Xu, et al: On General Binary Relation Baed Rough Set Thu, x ( XY) R XY ( X Y ) = ( X ) ( Y ) i proved. ( R X) ( R Y) ( x ( X )) ( x ( Y )) x ( X ) ( Y ). (4b) Since X X Y and YX Y, the formula ( X ) ( Y ) ( X Y ) i obviou from item (3b) in thi theorem. Moreover, we have that x ( XY) ( R XY) ( R = ) ( R X) ( R = ) ( R Y) [( R X) ( R = )] [( R Y) ( R = )] ( x ( X) ) ( x ( Y)) x ( X ) ( Y ). Hence, ( X Y ) ( X ) ( Y ) hold. Thi item i proved. tem (5a) (5b) can be proved directly and repectively by item (3a) (3b) in thi theorem. From Definition 5.1 and Theorem 5.1, it can be known that the greatet trength of GBRS() i that object preciely upport concept are one carrying poibly upport information with repect to the relation. That i to ay ( X ) ( X ). t accord with the key idea and the meaning of rough approximating to concept in rough et theory. Empty decription are conidered and dipoed a upporting any concept preciely and poibly upporting all concept with repect to the relation. The depiction to concept are more comprehenive and integrated by thi dipoal of empty information. But the lower and upper approximation operator in GBRS till can t atify duality and the dual ynchronou change of lower and upper approximation can't hold with repect to arbitrary general binary relation in GBRS(). Though the duality i not atified, GBRS() ha the ability to repreent concept approximately in rough et theory. The example developed in ection 3 will be till employed in thi ection to illutrate GBRS() a follow. Example 5.1 (Continued from Example 3.1) From Example 3.1, calculate the lower and upper approximation and clear that the duality doen t hold in GBRS(). ccording to Definition 5.1 and Example 3.1, the approximation are calculated and lited in the following. ( X ) = { x, x }; ( X1) = { x1, x3, x4, x5}; ( X ) = { x, x, x, x }; ( X ) = { x, x, x, x, x } Furthermore, X1= { x3, x5}, X2= { x1, x2}. Moreover, we have that ( X ) = { x, x }; ( X1) = { x2, x3, x4, x5}; ( X ) = { x }; 2 4 ( X ) = { x, x } Then, we can eaily verify that the duality doen t hold in GBRS(), either. Other propertie will be not verified in thi ection. 6. xiomatic GBRS From the above approache, we can ee that the lower and upper approximation operator in GBRS() JC for contribution: editor@jic.org.uk

10 Journal of nformation and Computing Science, Vol. 7 (2012) No. 1, pp atify duality but thoe in GBRS() and GBRS() don t atify the duality. Empty decription are only employed to depict concept preciely but not conidered a poibly upport decription in GBRS(). nlike GBRS(), empty decription are ignored totally in GBRS() and are employed to characterize concept both preciely and poibly in GBRS(). GBRS() i generalized and obtained by contructive method in rough et theory. Both the lower and upper approximation operator of GBRS() keep identical with thoe of Pawlak rough et in the form and expreion. The upper approximation enure the correction of decription poibly upport concept. But the lower approximation import extra object carrying empty information with repect to general binary relation. Thee object are included in lower approximation et but not included in upper approximation et. With empty decription being ignored and left out, GBRS() can depict concept more feaible and reaonable correponding to the rough approximation idea in rough et theory. empty decription being conidered in both lower and upper approximation, GBRS() ha the capability to depict concept with more comprehenive information. But the duality fail in GBRS() and GBRS(). The approximation operator can t act dually. ccording to the axiomatic method in rough et theory and the above three form of GBRS, lower and upper approximation operator atify duality can be advanced and dicued further. new form of GBRS can be defined and tudied in the following. Definition 6.1 Let be the univere. R i a general binary relation on the univere. For any X, the lower and upper approximation of X with repect to the generalized approximation pace are defined, repectively, a () ( X ) = { x ( R X ) ( R )}, ( X ) = { x ( R X ) ( R = )}. X i definable with repect to if and only if ( X ) = ( X ). Ele, X i rough with repect to if and only if ( X ) ( X ). Thi form of rough et i called the forth type of general binary relation baed rough et and denoted by GBRS(). Similarly a the above approache, propertie of lower and upper approximation operator in GBRS() will be dicued and tudied in the following theorem. Theorem 6.1 Let be the univere. R i a general binary relation on the univere. For any X, Y, the following propertie of lower and upper approximation operator with repect to generalized approximation pace hold in GBRS(). x ( X ) x ( X ) x ( X ) x X = X X = ar p X ); = = (1a) ( X ) ( X ); (1b) R X, R = iordered while ( X ) = ; (1c) X R, while R = ; (2a) ; (2b) ( (3a) ; (3b) ; Proof. (1a)From Definition 6.1, we can eaily have that x ( X ) ( R X ) ( R ) R X= R x ( X ). Hence, (1b) While ( X ) ( X ) i proved. ( X ) =, x ( X ) = = (4a) XY ( X ) ( Y); (4b) X Y ( X ) ( Y ); (5a) ( X Y ) ( X ) ( Y ); (5b) ( X Y ) ( X ) ( Y ); (6a) ( X Y ) ( X ) ( Y ); (6b) ( X Y ) ( X R = and thi item i obviou. ume that ) ( Y ). ( X ). For any JC for ubcription: publihing@w.org.uk

11 64 Weihua Xu, et al: On General Binary Relation Baed Rough Set x ( X ), one can obtain that ( R X) ( R ). Then, we have that x ( X) property i proved. (1c) From R =, one can have that { R x } contruct a covering of the univere. Thu, x Moreover, from Hence, y X x t.. y R y R X X { R R X }. x ( X ) ( R ( x ) X ) ( R ( x ) = ), one can obtain that R = { { R R X }} { { R R = }}. x X X { R R X } R. x x ( X ) R X. Hence, the Thi item i proved. (2a) i known R X R X=,thi item can be proved from Definition 6.1 a follow. ( X ) = { x ( R X ) ( R )} = { x ( R X= ) ( R )} = { x ( R X ) ( R = )} = ( X ). (2b) Thi item can be proved imilarly a item (2a) in thi theorem. ( X ) = { x ( R X ) ( R = )} = { x ( R X= ) ( R )} ={ x ( R X) ( R )} = ( X ). (3a) From the property of cantor et, there exit no object atifying ( R ) ( R ), i.e., ( )={ x ( R ) ( R )} =. Moreover, one can eaily obtain that ( )={ x ( R ( x ) ) ( R ( x )= )} = { x R ( x )= }. Hence, thi item i proved. tem (3b) can be proved by the duality or proved imilarly a item (3a) in thi theorem. (4a) Since X Y, we have that Thu, ( X ) ( Y ). Thi item i proved. x ( X ) ( R X ) ( R ) ( R Y) ( R ) x ( Y). tem (4b) can be proved imilarly a item (4a) in thi theorem. tem (5a)(5b)(6a)(6b) can be proved a thoe in the above ection. GBRS() i a more feaible model baed on general binary relation. The lower and upper approximation operator atify duality and hold the property lower approximation i included in the upper approximation. Empty decription are conidered a preciely upport none concept but poibly upport arbitrary one with repect to general binary relation in rough et theory. Thi dipoal of empty information can promote concept decription and repreentation more reaonable and logical. Knowledge acquiition can be more precie and integrated in problem conulting with general binary relation baed rough et. We till employ Example 3.1 to illutrate GBRS() a follow in thi ection. JC for contribution: editor@jic.org.uk

12 Journal of nformation and Computing Science, Vol. 7 (2012) No. 1, pp Example 6.1(Continued from Example 3.1) From Example 3.1, calculate the lower and upper approximation and clear that the duality doen t hold in GBRS(). ccording to Definition 6.1 and Example 3.1, the approximation are calculated and lited in the following. ( X ) = { x }; 1 1 ( X1) = { x1, x3, x4, x5}; ( X ) = { x, x, x }; ( X ) = { x, x, x, x, x } Furthermore, X1= { x3, x5}, X2= { x1, x2}. Moreover, we have that ( X ) = { x }; 1 2 ( X1) = { x2, x3, x4, x5}; ( X ) = ; 2 ( X ) = { x, x } Then, we can eaily verify the propertie in Theorem 6.1 and they will not be arranged in thi ection for being obviou. 7. Comparion and Relation among Four Type of GBRS defined and dicued eparately in the above ection, four type of GBRS can be related and compared. The following theorem can be concluded and tudied. Theorem 7.1 While the general binary relation i erial, the four type of GBRS defined by ucceor neighborhood are equivalent. Theorem 7.2 While the general binary relation i invere erial, the four type of GBRS defined by proceor neighborhood are equivalent. Theorem 7.3 While the general binary relation i reflexive, lower and upper approximation operator in the * four type of GBRS all atify * ( X ) X, where * repreent the type of GBRS (), (), (), (), repectively. Theorem 7.4 While the relation are equivalence relation, then four type of GBRS are all Pawlak rough et. Theorem 7.5 Let be the univere. R i a general binary relation on the univere. For any X, the following relation among four type of GBRS hold. = = = = (1) ( X ) ( X ) ( X ) ( X ); (2) ( X ) ( X ) ( X ) ( X ); (3) ( X ) ( X ); (4) ( X ) ( X ). Proof. Thee theorem are obviou and can be proved directly by the correponding definition and propertie. Other pecial general binary relation can be conidered and we don t dicu them any more in thi paper. n addition, empty decription and empty information are uually exited in many rough et model and they can be treated according to requirement imilarly a one of the four form preented in thi paper. 8. Concluion ccording to the theoretical contructive method and axiomatic method in rough et theory, general binary relation baed rough et wa dicued and tudied in thi paper. The key idea of approximative repreentation to concept and the form of information repreentation were conidered. Empty decription and empty information were dipoed in different way. By analyzing of the general binary relation baed rough et contructed from Pawlak rough et, three new type of general binary relation baed rough et were JC for ubcription: publihing@w.org.uk

13 66 Weihua Xu, et al: On General Binary Relation Baed Rough Set defined and tudied. Moreover, important propertie of different lower and upper approximation operator were dicued and approached in further. The four type of general binary relation baed rough et have valuable ignificance under different perpective and requirement. nd they can adapt the approximating to concept better in application and reearche employing general binary relation baed rough et. Propertie of different lower and upper approximation operator can promote the correponding type of general binary relation baed rough et more effective and uccinct whenever being ued. The four type of general binary relation baed rough et can be aociated with each other to improve the generalization ability and continuability of rough et in dealing with more practical problem. 9. cknowledgement Thi work i upported by Potdoctoral Science Foundation of China (No ) and National Natural Science Foundation of China (No , , and ). 10. Reference [1]. Maahiro and T. Tetuzo. Generalized rough et and rule extraction. in Rough Set and Current Trend in Computing. Berlin: Springer Berlin/Heidelberg. 2002, 2475: [2] K. Michiro. On the tructure of generalized rough et. nformation Science. 2006, 176: [3] Z. Pawlak and. Skowron. Rudiment of rough et. nformation Science. 2007, 177: [4] Z. Pawlak and. Skowron. Rough et: Some Extenion. nformation Science. 2007, 177: [5] Z. Pawlak and. Skowron. Rough et and Boolean Reaoning. nformation Science. 2007, 177: [6] Z. William and F.Y. Wang. Binary Relation Baed Rough Set. FSKD, LN. 2006, 4223: [7] Z. William. Relationhip between generalized rough et baed on binary relation and covering. nformation Science. 2009, 179: [8] Z. William. Generalized rough et baed on relation. nformation Science. 2007, 177: [9] W.H. Xu and W.X. Zhang. Meauring roughne of generalized rough et induced by a covering. Fuzzy Set and Sytem. 2007, 158: [10] W.H. Xu and W.X. Zhang. Method for knowledge reduction in inconitent ordered information ytem. Journal of pplied Mathematic and Computing. 2008, 26: [11] W.H. Xu, X.Y. Zhang, J.M Zhong and W.X. Zhang. ttribute reduction in ordered information ytem baed on evidence theory. Knowledge and nformation Sytem. 2010, 25: [12] Y.Y. Yao. Generalized rough et model. in Rough Set in Knowledge Dicovery. Polkowki and Skowron, Ed. Heidelberg: Phyica-Verlag. 1998, pp [13] Y.Y. Yao and T.Y. Lin. Generalization of rough et uing modal logic. ntelligent utomation and Soft Computing. 1996, 2: [14] W.X. Zhang, W.Z. Wu, J.Y. Liang and D.Y. Li. Rough Set Theory and Method. Beijing China: Science Pre, [15] W.X. Zhang, Y. Liang and W.Z. Wu. nformation Sytem and Knowledge Dicovery. Beijing, China: Science Pre, JC for contribution: editor@jic.org.uk