Generalized Infinitive Rough Sets Based on Reflexive Relations

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1 2012 IEEE International Conference on Granular Computing Generalized Infinitive Rough Sets Based on Reflexive Relations Yu-Ru Syau Department of Information Management National Formosa University Huwei 63201, Yunlin, Taiwan Jinyan Fan Department of Mathematics Shanghai JiaoTong University Shanghai , China Tsau Young Lin Department of Computer Science San Jose State University San Jose, CA , USA Abstract-We associate a binary neighborhood system with a given binary relation, then re-examine the structure of binary relational rough set model from the viewpoint of binary neighborhood systems. Based on the associated binary neighborhood system, we define an equivalence relation on the universe of discourse by regarding objects of the universe as equivalent if their associated binary neighborhood systems are identical. The induced equivalence relation is interpreted as the indiscernibility relation on the universe. From this view, we propose equivalent form of lower upper approximations in binary relational rough set model in ways paralleling the corresponding definitions for Pawlak's lower upper approximations. In addition, we extend variable precision rough set model to variable precision binary relational rough set model. Index Terms-rough set; binary relation; neighborhood; generalized approximation spaces; variable precision rough set model. 1. Introduction In rough set theory it is usually assumed [10], [11] that with every object of the universe of discourse U some information (data, knowledge) is associated, that the knowledge about objects is restricted by some indiscernibility relation which was assumed to be an equivalence relation. The R-equivalence classes are referred as R-elementary sets, or briefly, elementary sets. According to Pawlak [8], [9], any set X U can be described by elementary sets a pair of lower upper approximations. Some generalizations of the classical rough set model have been made by considering weaker forms of indiscernibility relation instead of an equivalence relation. For example, in [11], Slowinski Verpooten extended the rough set theory to the use of similarity relation (not necessarily symmetric or transitive) suggested that the reflexivity property seems quite necessary to express any form of indiscernibility or similarity, the two other properties (symmetry transitivity) may be relaxed. So elementary sets are extended to cover the neighborhoods of such binary relations. We shall call the knowledge represented by them elementary knowledge. Pawlak's lower upper approximations are indeed the interior closure for a topology on U. In this paper, we re-examine the topological structure of binary relational rough set model from the viewpoint of binary neighborhood systems [3], [4], [6]. Due to the known [3], [4], [6] fact that a binary relation is equivalent to a binary neighborhood system, we associate a binary neighborhood system with a given binary relation B U x U. Based on the associated neighborhood system, we define an equivalence relation on U by regarding objects of U as equivalent if their associated neighborhood systems are identical. The induced equivalence relation is interpreted as the indiscernibility relation on the universe of discourse U its equivalence classes are regarded as elementary sets. We present equivalent formulations, with the viewpoint of Artificial Intelligence (AI), of lower upper approximations in a binary relational rough set, extend variable precision rough set model to variable precision binary relational rough set model. II. Preliminary Let U be a certain nonempty set (may be finite or infinite) referred as the universe of discourse (in short the universe). That is, U is a fixed nonempty set containing all the objects under consideration. The power set of U, denoted by P(U), or 2u, is the collection of all subsets of U. That is, P(U) = {S I S U}. We sometimes express A B by writing B A, or by saying that B is a superset of A (or contains A). If A B A -I=- B, we write A c B or B A. Recall that [6] a collection A P(U) is said to be closed under supersets, or meet superset condition, if M N for some nonempty N E A, then ME A. Definition 1: A family T of subsets of U is called a topology on U if only if T contains the empty set o the whole space U, is closed under finite intersections arbitrary unions. The members of T are called open sets of (U, T). The complement of an open set is called a closed set. Topology can alternatively be described in terms of closed sets: The whole set the empty set are closed, arbitrary intersections of closed sets are closed, finite unions of closed sets are closed /12/$ IEEE

(TNS 2) If M N(x) for some nonempty N(x) E NS(x), then ME NS(x). (TNS 3) If N, ME NS(x), then N n ME NS(x).

2 A. Pre-/topological Neighborhood Systems Let us recall [1], [6]. Definition 2: For each x E U, let NS(x) = {N(x) U} be a nonempty family of subsets of U associated with x such that: (TNS 1) x E N(x) for each N(x) E NS(x). (TNS 2) If M N(x) for some nonempty N(x) E NS(x), then ME NS(x). (TNS 3) If N, ME NS(x), then N n ME NS(x). (TNS 4) If N E NS(x), then there exists M E NS(x) such that M N M E NS(y) for each y E M (that is, M is a neighborhood of each of its points). Then NS(x) is called a topological neighborhood system at x. We shall call N S (x) a pretopological neighbor hood system at x if it satisfies (TNS 1), (TNS 2), (TNS 3), but not necessarily (TNS 4). Remark 1: Axioms (TNS 1), (TNS 2), (TNS 3) (TNS 4) are referred as the axioms of topological neighbor hood system (in short TNS-axioms). Theorem 1: [1] TNS-axioms uniquely determines the topology T on U vice versa. B. Neighborhood Systems Binary Neighborhood Systems In [6], we defined a neighborhood system, or pretopology, on U as a mapping NS : U ---7 P(P(U)). For each x E U, N S (x) P (U) is called a neighbor hood system at x each member of N S (x) is called a neighborhood of x. A nonempty neighborhood N(x) E NS(x) is said to be anti-reflexive (punctured) if x 'f. N(x)'Vx ; reflexive (non-punctured) if N(x) contains x,'vx. We will be interested in neighborhood systems NS(U) = {NS(x) I x E U}, where NS(x) is a singleton { B (x)}. In this case, instead of singleton, we will use the element. So NS will be replaced by BN : U ---72u. C. Kuratowski Closure Axioms Let us recall the Kuratowski closure axioms (Kelley [1]). 4) For X, Y E P(U), c(x u Y) = (Preservation of binary unions). c(x) U c(y) Note that the fourth KC-axiom implies that if X Y U then c(x) c(y). Definition 4: A mapping c : P(U) --* P(U) satisfying the Kuratowski closure axioms is called a closure operator on U. If the axiom of Idempotence is relaxed, then the other three of the KC-axioms define a preclosure operator. D. Approximation in Binary Relations Let R U x U be a binary relation on U. We usually write xry to indicate that the ordered pair (x, y) is an element of R. R is said to be serial if for all x E U, there exists y E U such that xry. Every reflexive relation is serial. A relation R on U is a preorder if it is reflexive transitive, it is called an equivalence relation if it is reflexive, symmetric, transitive. If R is an equivalence relation on U, we say that objects x y are equivalent to indicate that xry. Definition 5: Let R be a binary relation on U. 1) For each x E U, we call the set BN(x) = {y lye U, xry} (11.1) the binary neighborhood of x with respect to R. 2) For any X U, the lower upper approximations, respectively, are defined as BN(X) = {x E U I BN(x) X} (II.2) BN(X) = {x E U I BN(x) n X =J 0}. (II.3) 3) The pair (U, R) is referred to as a binary relational (infinite) rough set model. Remark 2 Definition 5 has many hidden points [5]. 1) The interior closure defined here is different from our (Lin & Syau) paper in GrC 2011, where we required BN(X) = {x E U I BN(X) =J 0/\.BN(x) X} (II.4) 2) BN(U) = U BN(0) = 0. 3) For any subfamily {A,\ 1,\ E A} P(U), we have Definition 3: The Kuratowski closure axioms (KCaxioms) is referred to the following properties of a mapping c: P(U) --* P(U). 1) c(0) = 0 (Preservation of nullary unions). 2) For X E P(U), X c(x) (Extensivity). 3) For X E P(U), c(c(x)) = c(x) (Idempotence). BN( n A,\) = n '\EA '\EA BN( U A,\) = '\EA '\EA BN(A,\) U BN(A,\). (II.5) (II.6)

4) Viewing the lower upper approximations as mappings from P(U) to itself, they are mutually dual in the sense that BN(X) = U -BN(U- X), V X U. (II.

We write IXI for the number of points in a set X U. Let R be an equivalence relation on U, let R* = {El' E2,..., Em} be the partition of U determined by R.

For any X U, according to Pawlak [8], [9], we can describe X by elementary sets a pair of lower upper approximations, R(X) R(X), respectively, defined by R(X) = U{Ej E R* I Ej X} (II.

Equivalently, as noted in Pawlak [8], we can describe any set X by taking the topological approach as follows. E.(X) = {x E U I [X]R X} (II.

3 4) Viewing the lower upper approximations as mappings from P(U) to itself, they are mutually dual in the sense that BN(X) = U -BN(U- X), V X U. (II.7) 5) If R is further assumed to be serial, then BN(0) = 0 BN(U) = U. E. Variable Precision Rough Set Model We now assume that the universe U is a finite set. We write IXI for the number of points in a set X U. Let R be an equivalence relation on U, let R* = {El' E2,..., Em} be the partition of U determined by R. The equivalence class containing an object x E U will be denoted by [X]R' In the rough set theory, R equivalence classes are referred as R-elementary sets or, briefly, elementary sets. For any X U, according to Pawlak [8], [9], we can describe X by elementary sets a pair of lower upper approximations, R(X) R(X), respectively, defined by R(X) = U{Ej E R* I Ej X} (II.8) R(X) = U{Ej E R* I Ej n X i= 0}. (II.9) The definition of lower upper approximations given by Equations (II.8) (II.9) has AI meaning. Equivalently, as noted in Pawlak [8], we can describe any set X by taking the topological approach as follows. E.(X) = {x E U I [X]R X} (II.10) R(X) = {x E U I [X]R n X i= 0}. (II.11) For any X C U, according to Ziarko [13], the misclassification, or inclusion, error e( Ej, X) of Ej relative to a set X U, can be defined by IEjnxl _ e ( Ej,X ) -I-lEi!' For a given threshold (3 E [0,0.5), define E j (3 X if only if f e(ej, X) {3. We can also express Ej (3 X by saying that Ej is approximately included in X with error {3. Using " (3" instead of " ", Ziarko [13] extended the classical rough set model to the variable precision rough set model (VPRS) by replacing Pawlak's lower upper approximations with the {3-lower {3-upper approximations. Definition 6: [13] Let U be a finite universe let R be an equivalence relation on U. Let R* = {El' E2,..., Em} be the partition of U determined by R. For a given parameter (3 E [0,0.5), the {3-lower {3-upper approximations, (3 -(3 R (X) R (X), respectively, of a set X U are defined by R (3 (X) = U{Ej E R* I Ej (3 X} (II.12) R (3 (X) = U{Ej E R* I e(ej,x) < 1- {3}. (II.13) III. Topological Structure of Binary Relational Rough Sets We now assume that R U x U be reflexive. That is, for all x E U, x E BN(x). Then, from Equation (II.3), we obtain X BN(X), V X U. (III. 14) Then, by Item 1 of Remark 1 Equation (II.6), we conclude that the upper approximation operator BN is a preclosure. We summarize the results of this discussion in the following: Theorem 2: Let R U x U be reflexive. For x E U, let BN(x) = {y lye U xry} on U. Then the upper approximation operator BN : P(U) ---+ P(U) defined by BN(X) = {x E U I BN(x) nx i= 0} is a preclosure operator. That is, the operator BN P(U) ---+ P(U) satisfies the following statements. 1) BN(0) = 0. 2) For X E P(U), X BN(X). 3) For X, Y E P(U), BN(X U Y) = BN U BN(Y). Theorem 3: Let R U x U be reflexive transitive. For x E U, let BN(x) = {y lye U xry} on U. Then the upper approximation operator BN(X) = {x E U I BN(x) nx i= 0} is a closure operator. That is, BN : P(U) ---+ satisfies the Kuratowski closure axioms. P(U) Proof In view of Theorem 2, all that remains is to prove idempotence. For X = 0: Since BN(0) = 0, we also have BN(BN(0)) = 0 = BN(0). By virtue of the extensivity of BN, it suffices to show that for any nonempty X U,

BN(BN(X)) BN(X)}. (III.15) Let X U be nonempty. Then, by the extensivity of BN, both BN(X) BN(BN(X)) are also nonempty. Let x be an object in BN(BN(X)). By definition, BN(x) n BN(X) i- 0.

16) BN(y) n Xi- 0. (III. 17) Combining Equations (III.16) (III.16), we obtain which implies BN(x) n Xi- 0, BN(BN(X)) BN(X)}.

approximation induced by a reflexive transitive relation is not idempotent obeys the four Kuratowski closure axioms.

4 BN(BN(X)) BN(X)}. (III.15) Let X U be nonempty. Then, by the extensivity of BN, both BN(X) BN(BN(X)) are also nonempty. Let x be an object in BN(BN(X)). By definition, BN(x) n BN(X) i- 0. Let y be an object in BN(x) n BN(X), so that y E BN(x) y E BN(X). From y E BN(x), since R is reflexive, we obtain at once that BN(y) BN(x)} BN(y) i- 0. On the other h, y E BN(X) implies (III.16) BN(y) n Xi- 0. (III. 17) Combining Equations (III.16) (III.16), we obtain which implies BN(x) n Xi- 0, BN(BN(X)) BN(X)}. Theorems 2 3 state that the upper approximation induced by a reflexive but not transitive relation is not idempotent obeys only three of the four Kuratowski closure axioms, that the upper approximation induced by a reflexive transitive relation is not idempotent obeys the four Kuratowski closure axioms. As will be shown in the following subsection, the binary neighborhoods of a reflexive transitive relation are open relatively to the topology determined by the upper approximation induced by the relation. This property is laking in a reflexive but not transitive relation. Note that for a reflexive but not transitive relation, the neighborhood system generated by the binary neighborhood system by superset condition is a pretopogical neighborhood system, that for a reflexive transitive relation, the neighborhood system generated by the binary neighborhood system by superset condition is a topological neighborhood system. A. Illustrative Examples The following examples illustrates Theorems 2 3. Example 1: Consider the set U = the universe of discourse. Let {Xl, X2, X3} to be Then R is reflexive but not symmetric or transitive. The binary neighborhoods are BN(xI) = BN(X2) = BN(X3) = The upper approximations are BN(0) = 0 BN({xd) = {Xl} BN({X2}) = {X2,X3} BN({X3}) = {XI,X3} BN({XI,X2}) = {XI,X2,X3} BN({XI,X3}) = {XI,X3} BN({X2,X3}) = {XI,X3} {X2} {X2,X3}. {XI,X2,X3} B N ( = { Xl, X2, X3}) {Xl, X2, X3}} 1) BN is not idempotent since, for example, BN({X2}) = {X2,X3},BN({X2,X3}) = but BN(BN({X2})) i- BN({X2}). 2) The family of complements of members of is a topology for U. Note that the binary neighborhood is not open. Example 2: Consider the set U = universe of discourse. Let R = {XI,X2,X3}, {Xl, X2, xd to be the {(Xl, xi), (X2, X2), (X3, X3), (Xl, X2), (X2, X3), (XI,X3)}. Then R is reflexive transitive. The binary neighborhoods are BN(xI) = {XI,X2,X3} BN(X2) = {X2, X3} BN(X3) = {X3}. The upper approximations are BN(0) = 0 BN({xd) = {Xl} BN({X2}) = {XI,X2} BN({xd) = {XI,X2,xd BN({XI,X2}) = {XI,X2} BN({XI,X3}) = BN({X2,X3}) = B N ( { Xl, X2, X3}) = {XI,X2,X3} {XI,X2,X3} {Xl, X2, X3}}

5 The family of complements of members of {0,{Xl},{Xl, X2}, {Xl, X2, X3}} F = is a topology for U, the binary BN(xI) {XI, X2, X3}, BN(X2) BN(X3) = {X3} are all open. neighbor hoods {X2,X3}, IV. Variable Precision Binary Relational Rough Set Model We associate an equivalence relation with a given binary neighborhood system as follows. Definition 7: Let R U x U be a binary relation on U. For X E U, let BN(x) = {y lye U, xry}. 1) The binary neighborhood system BN(U) = {BN(x) I X E U} thus generated will be called a binary neighborhood system associated with R. 2) Define an equivalence relation J(BN) on U by regarding objects x, y of U as equivalent if BN(x) = BN(y). The equivalence class containing an object X E U will be denoted by [Xh(BN)' Using Definition 7 Equations (II.4) (II.3), we immediately obtain the following: Lemma 1: Let R U x U be serial. If (x, y) E J(BN), then for any X U, 1) X E BN(X) if only if y E BN(X). 2) X E BN(X) if only if y E BN(X). Using Lemma 1 Equations (11.4) (11.3), we immediately obtain the following: Theorem 4: Let R U x U be serial. For X E U, let BN(x) = {y lye U, xry}. By J(BN) we mean the equivalence relation on U by regarding objects x, y of U as equivalent if BN(x) = BN(y), [Xh(BN) denotes the equivalence class containing an object X E U. Then the lower upper approximations can be also described by J(BN)-elementary sets as shown below: BN(X) = U{[Xh(BN) I [X]I(BN) X} BN(X) = U{[Xh(BN) I [Xh(BN) n X -I- 0}. (IV.lS) (IV.19) Using " r/ ' instead of " ", we extend Ziarko's variable precision rough set model to variable precision binary relational rough set model by replacing Ziarko' ;3-lower ;3-upper approximations with BN f3 -lower BN f3 -upper approximations, respectively, as shown below: Theorem 5: Let R U x U be serial. For X E U, let BN(x) = {y lye U, xry}. By J(BN) we mean the equivalence relation on U by regarding objects x, y of U as equivalent if BN(x) = BN(y), [X]I(BN) denotes the equivalence class containing an object X E U. For a given parameter ;3 E [0,0.5), the BN f3 -lower BN f3 -upper approximations, respectively, of a set X U are defined by BN f3 (X) = U{[Xh(BN) I [Xh(BN) f3 X} (IV.20) -f3 BN (X) = U{[X]I(BN) I e([xh(bn), X) < 1-;3}. (IV.21) V. Conclusion One of the basic idea of rough set theory is the assumption that objects characterized by the same information are indiscernible in view of the available information about them. Any set of all indiscernible objects is referred as an elementary set forms a basic granule of knowledge about the universe. Pawlak introduced two definitions for the lower upper approximations with reference to an indiscernibility relation which was assumed to be an equivalence relation. One has AI meaning, the other is formulated by taking the topological approach. The definition of lower upper approximations defined by expressing the lower approximation as the union of all elementary sets which are contained in the subset of interest the upper approximation as the union of all elementary sets, which have a nonempty intersection with the set has AI meaning. In this paper, we re-examine the topological structure of binary relational rough set model from the viewpoint of binary neighborhood systems. We associate a binary neighborhood system with a given binary relation B U x U, then define an equivalence relation on the universe by regarding objects of the universe as equivalent if their associated neighborhood systems are identical. The induced equivalence relation is interpreted as the indiscernibility relation on the universe its equivalence classes are regarded as elementary sets. Consequently, in a binary relational rough set model, the lower approximation of a subset of interest X U is the union of all the elementary sets which are contained in the subset of interest, whereas the upper approximation of a set X is the union of all the basic granules which have a nonempty intersection with the subset. In this way, equivalent form, with AI view, of lower upper approximations in a binary relational rough set model is formulated. In

6 addition, based on the equivalent form, we extend variable precision rough set model to variable precision binary relational rough set model. Acknow ledgment The first author would like to thank the National Science Council of Republic of China for financial support through NSC loo-2221-e References [1] J. L. Kelley, General Topology, Van Nostr, Princeton, N. J., [2] M. Kondo, On the structure of generalized rough sets, Information Sciences 176 (5) (2005) [3] T. Y. Lin, Granular Computing on Binary Relations I: Data Mining Neighborhood Systems. In: Rough Sets Knowledge Discovery, A. Skowron L. Polkowski (eds), Physica -Verlag, 1998, [4] T.Y. Lin, Granular computing on binary relations II: rough set representation belief function, in: A. Skowron L. Polkowski (Eds), Rough Sets Knowledge Discovery, Physica -Verlag, 1998, pp [5] T.Y. Lin, Granular Computing for Binary Relations: Clustering Axiomatic Granular Operators. In: Proceedings of the 2004 North American Fuzzy Information Processing Society Annual Conference, Banff, Alberta, Canada, June 27-30, 2004, pp [6] T. Y. Lin, Y. R. Syau, Keynote Speech: Granular Mathematics - Foundation Current State. A Keynote Speech. In: Proceedings of the 2011 IEEE International Conference on Granular Computing, Kaohsiung, Taiwan, November 8-lO, 2011, pp [7] G. Liu W. Zhu, The algebraic structures of generalized rough set theory, Information Sciences 178 (21) (2008) 4lO [8] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, [9] Z. Pawlak, Rough sets, International Journal of Computer Information Science 11 (1982) [lo] Z. Pawlak, A. Skowron, Rough sets: some extentions, Information Sciences 177 (1) (2007) [11] R. Slowinski D. Verpooten, A Generalized definition of rough approximations based on similarity, IEEE Transactions on Data Knowledge Engineering 12 (2000) [12] H. P.,Zhang, Y, Ouyang, Z.,Wang, Note on "Generalized rough sets based on reflexive transitive relations". Information Sciences 179 (2009) [13] W. Ziarko, Variable precision rough set model, Journal of Computer System Sciences 46 (1) (1993)

On Generalizing Rough Set Theory

On Generalizing Rough Set Theory Y.Y. Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca Abstract. This paper summarizes various formulations