2 T.Y. Lin Acknowledgement This research is partially supported by Electric Power Research Institute, Palo Alto, California, San Jose State University

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1 Granular Computing on Binary Relations II Rough Set Representations and Belief Functions T.Y. Lin Department of Mathematics and Computer Science San Jose State University San Jose, California And Department of Electric Engineering and Computer Science University of California Berkeley, California or 1 Introduction This is a continuation of [13]. Let us quote few words from it. "Granulation :::appears :::in dierent names, such as chunking, clustering, data compression, divide and conquer, information hiding, interval computations, and rough set theory, just to name a few." " the computing theory on information granulation has not been fully explored in its own right." ":::in mathematics, the notion of partitions is well explored." According to Lot Zadeh [38], { "information granulation involves partitioning a class of objects(points) into granules, with a granule being a clump of objects (points) which are drawn together by indistinguishability, similarity or functionality." By observing some technical points [13], we rephrase his words as follows: { information granulation is a collection of granules, with a granule being a clump of objects (points) which are drawn towards an object. In other words, each objects is associated a family of clumps. Mathematically the association of each object with such a family of clumps is a neighborhood system [12, 13]. If all clumps of a neighborhood system are nonempty, then the neighborhood system is a covering. If the family of all clumps forms a partition, the neighborhood system reduces to rough set system. If we restrict ourselves to real numbers, then the computing of neighborhood systems is reduced to interval computing. Intuitively neighborhoods are a common notion, it has been used in data mining, intelligent query, soft computing, logic [1, 35, 9, 10, 3,2,11,5,20,21]More recent studies have appeared in [12, 36,13]. In Part I [13], we focus on the structure theory, in this second part, we turn to representations theory. Rough set methodology (information table processing) is extended to binary relations (neighborhood systems).

2 2 T.Y. Lin Acknowledgement This research is partially supported by Electric Power Research Institute, Palo Alto, California, San Jose State University, NASA Grant NCC2-275,ONR Grant N , LLNL Grant , ARO Grant DAAH , and BISC Program of UC Berkeley. 2 Binary Relations and Neighborhood Systems Let U be a crisp set. Let B V U be a binary relation. For each object p 2 V, we associate a binary subset B p U, where B p = fu j pbug that consists of all elements u that are related to p by B. The map B : p ;! B p or the collection fb p g is referred to as a binary neighborhood system. Wehave shown that a binary relation induces a binary neighborhood system. It is easy to see the converse is also true. It is also clear how one can formulate the fuzzy case we will skip the details. In many applications, we need to consider several clumps at the same time, so we dene A neighborhood system is a map NS : p 2 V ;! fn p g2 2U that associate each object p a family of clumps (crisp/fuzzy subsets), which may be empty, nite or innite. These clumps are called fundamental neighborhoods. In rough set system such a clump is an elementary set (equivalence class). In binary relation, it is an elementary neighborhood. V is called an object space, U a data space. The set of all the names of clumps is called a concept space. The 4-tupe (V U B C) is called a binary granular structure. A partition is a collection of pair-wise disjoint subsets whose union is U. If B is an equivalence relation, then fb p g forms a partition. Note that B p =B q,if p is equivalent toq. Rough sets form a special form of neighborhood system we will call it a rough set system. In this paper, rough set will be used in this sense. Partition is the neatest kind of granulation. A covering is a collection of non-empty subsets fx j g(may notbe disjoint) such that U = S fx j g. X j is called a cover. It is a "common" geometric generalization of partition, however, there is no algebraic counter part. We should like to note that a covering is a (general) neighborhood system: X j is a fundamental neighborhood of every p 2 X j. Note that p may becovered by several covers, so fx j g is not a binary neighborhood system. Intuitively, an elementary neighborhood B p is more than a cover,itisa"cover with a center at p." The notion of

3 Rough Sets and Belief Functions 3 a "center at p" plays an essential role in providing the mapping from an object to the cover with a "center at this object". Such a map is the key for extending rough set methodology to general binary relations see Section 4. Let us recall some specic binary neighborhood systems 1. B is serial, if 8p B p is non-empty, 2. B is reexive, if 8p p 2 B p 3. B is symmetric, if 8p 8q q 2 B p =) p 2 B q 4. B is transitive, if 8p 8q 8r q2 B p and r 2 B q =) r 2 B p 5. B is Euclidean, if q 2 B p,andr 2 B p =) r 2 B q 6. B is rough set (clopen), if it is reexive, symmetric, and transitive. 3 Word Representations of Rough Sets Zdzislaw Pawlak regards classication (equivalence relations) as knowledge. He calls a universe with n classications a knowledge base and shows that a knowledge base can be represented by an information table and vice versa [23]. Proposition. There is an one-to-one correspondence between Pawlak knowledge bases and information tables. 3.1 Information and Decision Tables An information table is a 4-tuple where (U T Dom ), 1. U = fu v :::g is a set of entities. 2. T is a set of attributes fa1 A2 :::A n g. 3. Dom(A i )isthesetofvalues of attribute A i Dom = dom(a1) dom(a2) ::: dom(a n ) 4. : U T ;! Dom, called description function, is a map such that (u A i )isindom(a i )foralluin U and A i in T. The description function induces a set of maps t = (u ;) :T ;! Dom. Each image forms a tuple: t =((u A1) (u A2) :::: (u A i ) ::(u A n ))

4 4 T.Y. Lin Note that the tuple t is associated with object u, but not necessarily uniquely. In an information table, two distinct objects could have the same tuple representation that is not permissible in relational databases. A decision table is an information table (U T Dom ) inwhich the attribute set T = C [ D is a union of two non-empty sets, C and D, of attributes. The elements in C are called conditional attributes. The elements in D are called decision attributes. Each row is a decision rule. 3.2 A Rough Set Example Recall that in rough set theory equivalence classes are called elementary sets. The central notion here is to give a name to each elementary set [11]. These names will be referred to as elementary concepts. Let U = fid1 ID2 :::ID9g be a set of 9 balls. Assume U can be partitioned into equivalence classes by their colors and weights. We name these colorelementary sets, Red, Orange, and Yellow and weight-elementary sets, W1, W2, W3, andw4 see Table 1 and 2. Elementary Concept Red Orange Yellow Elementary Set ID1 ID2 ID3 ID4 ID5 ID6 ID7 ID8 ID9 Table 1. Color Partitions and Their Names Elementary Concept W1 W2 W3 W4 Elementary Set ID1 ID2 ID3 ID4 ID5 ID6 ID7 ID8 ID9 Table 2. Weights Partitions and Their Names In terms of these names, we have an information table that represents the knowledge base (U, color, weight) see Table 3. This is how rough set theory represents equivalence relations into an information table. If the family of equivalence relations under consideration forms a nested sequence, we do not use information tables, we often represents the family in the

5 Rough Sets and Belief Functions 5 Balls Colors Weight ID1 Red W1 ID2 Red W1 ID3 Red W2 ID4 Orange W3 ID5 Orange W3 ID6 Yellow W4 ID7 Yellow W4 ID8 Yellow W4 ID9 Yellow W4 Table 3. A Representation by Colors and Weights form of tree, called concept hierarchy tree. Such tree representations can also be extended to binary relations [15]. 4 Representations of Binary Relations Real world granulation often can not be expressed by equivalence relations. For example, the notions of "near," "similar," "enemy of," and "conict" are not equivalence relations. So there are intrinsic needs to generalize the theory of equivalence relations (rough set theory) to the theory of general binary relations (binary neighborhood systems). Since information table processing is very useful, in this section we will represent binary relations in table format. The table will be called an extended information table. Through such representations, information table processing(rough set methodology) is extended to process granular structures of (general) binary relations. It seems paradoxical that we can have table and tree representations for general binary relations. The key to such a representation is the existence of the mapping B : V ;! 2 U : p ;! B p that maps an object to its unique binary neighborhood (the unique "cover with a center") and hence to its unique name (elementary concept). If the object space V is a nite set, the map can be represented by a table format. We will call it extended information table see Section 4.1. We should caution readers that these binary neighborhoods are, in general, not disjoint and hence their names are, in general, semantically not independent. We should also emphasize that, in contrast to the information table and equivalence relations, extended information table is not a perfect representation of binary relations. In other words, extended information table only reect some partial properties of the set of binary relations. For example, the intersections of neighborhoods (overlapping semantics) are not represented. We will discuss this situation in

6 6 T.Y. Lin the next paper [14]. Perhaps we should also point out that this representation induces a "new" classication on V see Section Extended Information tables We will illustrate such a generalization by modifying the last example. First let us note that each color is a combination of one or more primitive colors. For example, orange consists of two primitive colors, red and yellow. Let us consider the following binary relation R. We write p R q, if the colors of ball p and q have a common primitive color. So we have a list of binary related balls for the ball ID1: ID1 R ID1 ID1 R ID2 ID1 R ID3 ID1 R ID4 ID1 R ID5. Each R related pair has one common primitive color, namely, red. We can make another list for ID2, and so on. However as we have remarked before, binary relation can also be expressed by binary neighborhood system, which is often more concise. So we will do so. Each ballp will be associated with a set B p of balls, in which each ball contains a primitive color of ball p. B ID1 = fid1 ID2 ID3 ID4 ID5g B ID2 = fid1 ID2 ID3 ID4 ID5g B ID3 = fid1 ID2 ID3 ID4 ID5g B ID4 = fid1 ID2 ID3 ID4 ID5 ID6 ID7 ID8 ID9g B ID5 = fid1 ID2 ID3 ID4 ID5 ID6 ID7 ID8 ID9g B ID6 = fid4 ID5 ID6 ID7 ID8 ID9g B ID7 = fid4 ID5 ID6 ID7 ID8 ID9g B ID8 = fid4 ID5 ID6 ID7 ID8 ID9g B ID9 = fid4 ID5 ID6 ID7 ID8 ID9g These 9 binary neighborhoods constitute 3 distinct sets. Their names are: Have R is the name for B ID1 = B ID2 = B ID3 these balls contain red primitive color. Have R Y is the name for B ID4 = B ID5 these balls contain red and yellow primitive colors. Have Y is the name for B ID6 = B ID7 B ID8 = B ID9 these balls contain yellow primitive color Example Assume U has another granulation as expressed in Table 2. Then we can form a Table 4. Perhaps, we should stress again that those words (name of elementary neighborhoods) in Table 4 have overlapping semantics.

7 Rough Sets and Belief Functions 7 BALLs Granulation 1 Granulation 2 ID1 Have R W1 ID2 Have R W1 ID3 Have R W2 ID4 Have R Y W3 ID5 Have R Y W3 ID2 Have Y W4 ID3 Have Y W4 ID4 Have Y W4 Table 4. A Extended Information Table for Binary Relations 4.2 Derived Partitions of Binary Relations From Table 4, it is clear that the map B : ID j ;! B IDj induces a cover B IDj with "center at ID j "j = 1 2 :::. Of course, there are, in general, overlapping among B IDj, so they do not form a partition on U. Onthe other hands, the complete inverse image of B form a new partition on V. This new partition is really the one that classies the objects ID j j = 1 2 :::. Recall that each cover has a name. The complete inverse images of distinct names form a covering of V. In notations, the complete inverse images of the three names, Have R Have R Y Have Y,are fid1 ID2 ID3g fid4 ID5g fid6 ID7 ID8 ID9g. They form a partition on V. There are applications for such derived partitions [35, 13]. We believe this derived partition of V is the partition considered by various authors for their generalized rough set models [24, 30, 31, 8, 32, 33]. Syntactically, the information table of the derived partition is isomorphic to the extended information table however, semantically, theyarenot isomorphic. 4.3 A Fuzzy Example Each real number is associated with a special fuzzy set F, called a fuzzy number [39]. We will denote such fuzzy numbers by fuzzy number(). We can regard F, as a fuzzy binary neighborhood of, and fuzzy number() as the name of the neighborhood. In notation, ;! F ;! fuzzy number() The real number is in the object space, F is a fuzzy subset of data space, and fuzzy number() is in the concept space. This is a sinlge-valued representation of real numbers in terms of the names of fuzzy sets.

8 8 T.Y. Lin 4.4 Multi-Valued Representations We could also take neighborhoods as covering, then some objects may belong to more than one elementary concepts. For example, if we take the fuzzy example in last section. Each real number is covered by innitely many fuzzy numbers (they are fuzzy sets). So the representation is multi-valued eachnumber is represented by innitely many fuzzy numbers. Of course, if the data space is a nite set, then the representation is a nitely many valued representation. We will discuss such representations in the next paper. 5 Extending Rough Set Methodology In last section, we extend Pawlak's knowledge representation (information table) of equivalence relations to that of binary relations. In this section, we will extend the rough set methodology (table processing) to binary relations. We establish many analogous results In [23], Pawlak develop a theory of knowledge based on equivalence relations. We will extend his idea to binary relations. We were tempted to use a knowledge-oriented terminology to express our results, which will be very appealing intuitively. However, our results are analogous to, but not exactly the same as Pawlak's, to avoid confusing, we willkeep mathematical terms dierent. The knowledge oriented terms will only be used for enhancing the intuition. Knowledge Pawlak Lin's More general oriented terms terms proposal proposal knowledge partition binary neighborhood neighborhood (geometric) (classication) system system knowledge equivalence binary (algebraic) relation relation granule elementary set elementary fundamental (equivalence class) neighborhood neighborhood concept elementary elementary fundamental space concept space concept space concept space knowledge Pawlak binary multivalued base knowledge base knowledge base knowledge base knowledge information extended multivalued Representation table information table information table Table 5. Samples of Corresponding Terms

9 Rough Sets and Belief Functions Binary Knowledge Bases For convenience, we will call a set of binary relations on nite universe a binary knowledge base. Let B be a binary relation or a binary neighborhood system. Recall that each point p 2 V has a unique neighborhood, called the elementary neighborhood of p. We will denote it by B-neighborhood of p. A subset X is called a denable B-neighborhood, if X is a union of elementary B-neighborhoods and a denable B-neighborhood of p if further the union contains the binary B-neighborhood of p. The set of all denable B-neighborhoods at p is denoted by BS(p). The set of all denable B-neighborhoods is denoted by BS(U). Let D be another binary relation. Then we have the following denition. Denition. 1. Knowledge D strongly depends on knowledge B, denoted by B =) D, i every binary D-neighborhood is a denable B-neighborhood. 2. If B =) D, wewillsay B is denably ner than D or D is denably coarser than B. 3. Knowledge D weakly depends on knowledge B, denoted by B ;! D, i every B-neighborhood is contained in a D-neighborhood. 4. If B ;! D, we will say B is weakly ner than D or D is weakly coarser than B. In rough set theory, weakly and strongly dependencies are the same. Example Let U be a Euclidean plane. Let X p (K) be a disk of radius K. Let us consider two neighborhood systems or coverings fx p (K)g K = 1 2. It is clear that every X p (1) is contained in a X p (2), However, fx p (1)g does not =) fx p (2)g. We will write D-neighborhood at p by D p. Let Y p = NAME(D p ) and X pi = NAME(B pj ) be meaningful names. Since D p = S B pj 's for suitable choices of p j 2 V,we will write informally Y p = X p1 S Xp2 S :::... Note that Y p and X pj are words. So, this "formula" is a computing with words. Proposition If B =) D, then there is an algebraic map from the concept space of B to the concept space of D in the following sense that the map can be expressed by Y p = X p1 S Xp2 S ::: Again we should caution readers that the obvious "extension" may not be true. This proposition is signicant, since NAME(B pj ) is not semantically independent. It implies that the semantic constraints among these words NAME(B pj ) are carried over to those words, NAME(D p )'s consistently. In applications, these names should be chosen carefully so that the overlapping semantics is readable in the linguistic forms, especially, when human interactions are expected.

10 10 T.Y. Lin Such semantic consistency among words of the same columns allows us to extend the operations of information table (of equivalence relations) to extended information tables (of binary relations). Next we turn to weakly dependencies. Proposition B \ D ;! D and B \ D ;! B, where \ denotes the intersection. Let T = fc1 C2 :::C m g be a collections of binary relations. We will write INT(T )=C1 \ C2 \ ::: \ C m. Denition: 1. B is dispensable in T if INT(T )=INT(T ; B) otherwise B is indispensable. 2. T is independent ifeach B 2 T is indispensable otherwise T is dependent. 3. S is a Reduct of T if S is a subset of T such that INT(S) =INT(T ). 4. The set of all indispensable relations in T is called a Core, and denoted by CORE(T ). 5. CORE(T )= T RED(T ), where RED is the set of all reducts in T. The fundamental procedures in information table processing are to nd cores and reducts of decision tables. We hope readers are convinced that we have developed enough notions to extend these operations to extended tables. More details will be reported in another paper in the near future. 6 Belief Functions In this section, we will examine the interactions of Dempster-Shafer theory of evidence with neighborhood systems and rough sets. Let us review a few basic terms. Denitions Let X be a subset of U. 1. The interior of X:I[X] =fp : 9 N p Xg i.e., I[X] is the largest open set contained in X. 2. The closure of X, C[X] =fp : 8N p X \ N p 6= g i.e., C[X] is the smallest closed set contains X. 3. I[X] and C[X] are precisely the lower and upper approximation, if the neighborhood system is a rough set system. 4. A collection of subsets of U is called a -algebra, if it is closed under complement, countable unions and intersections [7]. 5. A triple (U P r) is called a probability space, if U is a set, is a -algebra, and Pr is a measure such thatpr(u) =1. Perhaps, we shouldstressthatwetake probability theory as an axiomatic mathematical system. So it is not important how those numerical values are assigned

11 Rough Sets and Belief Functions 11 (subjectively, objectively, or others). As long as the operations of probability obey the axioms, we will call them probability. According to [25], a frame of discernment is a nite set. To be consistent with other sections, we will deviate from tradition and refer to it as the universe. For a universe U, a function m P : 2 U ;! [0 1] is called a basic probability assignment(bpa) if m( ) = 0 and AU m(a) =1.Any A U such thatm(a) 6= 0 is called a focal element of m. Given a bpa m, the belief and plausibility functions over U, Bel Pl :2 U ;! [0 1], are dened by the following equations, for all A U, Bel(A) = X BA m(b) and Pl(A) = X B\A6= m(b): The dual relation Pl(A) = 1; Bel(A) holds between these two functions. 6.1 Belief Functions - A Counter Intuitive Measure One of the most counter intuitive part of belief functions is its denition. A belief function takes a simple minded direct sum of the measures of focal elements as its value. It computes the belief function as if there are no intersections. As a consequence, the belief measures of pure intersections are always 0. An intersection is said to be pure, if it does not contain any other focal element. Example 1 A counter intuitive example Let the universe be U = f g, andtwo focal elements be X = f g and Y = f g. Let their bpa's be m(x) =2=3 andm(y )=1=3. We set rest of bpa's to zero. Then by denition, Bel(X) =2=3 andbel(y )=1=3 Next let us compute the belief measure of the intersection, Bel(X \ Y )=0. As expected, the belief measure of pure intersection is zero. It seems paradoxical that intuitively the Dempster-Shafer's evidence of any two pure events are always "disjoint" (extending the term "pure intersection" from focal elements to events). We will present several interpretations to this problem.

12 12 T.Y. Lin 6.2 From Rough Set Theory Rough set theory is a theory of partitions all equivalence classes are disjoints. So rough setters often consider only the cases where focal elements are mutually disjoint. Such a situation arise very naturally in probability theory. Let a bpa m be given, and assume the family of the focal elements F = ff1 F2 :: F m g forms a partition. Then a natural probability and a belief function can be dened. We will discuss some interesting interaction between them. Note that the -algebra generated by F is the family of denable sets, DEF(F ). Next, dene Pr(F j ) = m(f j ) and extend it to DEF(F ) additively. Then the triple (U DEF (F ) Pr) forms a probability space. By computing Bel, Pl, and Pr,wehave Proposition. Bel(A) = Pr(I[A]), and Pl(A) = Pr(C[A]). There are many concrete interpretations and variants of this result in rough set theory (e.g.,[22, 27, 28, 29, 6]). For example, if we take m = Card(F j )=Card(U), whre Card(X) is the cardinal number of X, then we have many concrete results. Some of these will be generalized see Section 6.4and 6.3. Example 2 A random variable Let Pr be a probability onu = f g dened by Pr(1) = Pr(2) = Pr(3) = Pr(4) = Pr(5) = Pr(6) = 1=6, Pr(7) = Pr(8) = 0 Assume we are given a random variable (a real valued measurable function)as follows: (1) = (2) = (3) = (4) = a (5) = b = (6) = b (7) = c = (8) = c where a, b and c are distinct real numbers. Then the inverse images are: ;1 (a) =X = f g ;1 (b) =Y = f5 6g ;1 (c) =Z = f7 8g By the additive property of probability, we can easily compute Pr(X) =Pr(1) + Pr(2) + Pr(3) + Pr(4) = 1=6+1=6+1=6+1=6=2=3 and Pr(Y ) = Pr(5) + Pr(6) = 1=6+ 1=6 = 1=3. Pr(Z) =0

13 Rough Sets and Belief Functions 13 Note that F = fx Y Zg is a partition. The -algebra generated by X, Y, Z is the family of denable sets, DEF(F ). It is easy to verify that (U DEF (F ) Pr) forms a probability space. Let us turn around and dene the bpa's of F by their probabilities. m(x) =Pr(X) m(y)=pr(y), and m(z) =Pr(Z) =0. We set rest of bpa's to zero. So X and Y are the only focal elements. Let A = X \ Y, then A = I[A] =. Let B = f g, theni[b] = Y [ Z. By computing Bel(;) and PR(I[;]) according to their respective definitions, we get the following: Bel(A) =Pr(I[A]) = 0 Bel(B) =Pr(I[B]) = 1=3 This example veries the last proposition. 6.3 From Probabilistic Multivalued Random Variables In stead of assuming focal elements are disjoint. We will use the notion of Probabilistic Multivalued Random Variables (PMRV) to construct focal elements with "disjoint measures." From Example 2, we see that a random variable denes a new probability space on U. In this section we will show a similar idea can, with more elaborate eorts, be extended to PMRV. Roughly, the bpa of a focal element is a weighted sum of the measures of individual points in the focal element. A point in a pure intersection distributes its measure to all focal elements according to the weights of a "multivalued" distribution. So intuitively the set of bpa's representsasetof "disjoint measures" of focal elements see [19, 18]. We will illustrate the idea by modifying Example 2. Example 3 A probabilistic multivalued random variable Let Pr be the probability dened in Example 2. Next, let us consider the following PMRV (a real multivalued measurable function): (1) = a (2) = (3) = (4) = (5) = fa bg w(a) =3=4 andw(b) =1=4 areweights. (6) = b (7) = (8) = c Their "inverse images" are: ;1 (a) =X = f g ;1 (b) =Y = f g ;1 (c) =Z = f7 8g The collection fx Y Zg does not form a partition, it is merely a granulation (a neighborhood system). We would like to make some "unconventional observation." We will take X as a "fuzzy set." Its membership function, denoted by F X, is dened by theweight w of PMRV asfollows:

14 14 T.Y. Lin F X (1) = 1, since 1 is not shared. F X (2) = F X (3) = F X (4) = F X (5) = w(a), since their memberships are shared with Y. F X (6) = F X (7) = F X (8) = 0, since they are not in X at all Similarly, F Y is dened by F Y (1) = 0, since 0 is not in Y. F Y (2) = F Y (3) = F Y (4) = F Y (5) = w(b), since they are shared with X. F Y (6)=1,sineitisfullyinY F Y (7) = F Y (8) = 0, since they are not in Y at all By the same spirit, we observe thatf Z is the empty fuzzy set, because F Y (1) = F Y (2) = F Y (3) = F Y (4) = F Y (5) = F Y (7) = F Y (8)=0 Intuitively, wewould say F X and F Y are "disjoint as fuzzy sets." The rational is as follows: For each member in the intersection, say 2, theweighted membership is w(b) for "2 2 Y " and w(a) for "2 2 X". Since w(a) +w(b) = 1, it implies that "2" has a full membership when X and Y are combined together, and none when we look at the intersection. In other words, 75% of "2" belongs to X, 25% of "2" to X, and 0% to the intersection (w(b)=3/4, w(a)=1/4). So F X F Y are "disjoint fuzzy sets" as we observed. Next wewould like to propose a new fuzzy measure NfmsuchthatNfm(F W ) is the bpa of W. In other words, we try to give each X, Y, or Z a mutually "disjoint bpa." The new fuzzy measure is a weighted sum of the probability measures. Nfm(F X )=Pr(1) + w(a)[pr(2) + Pr(3) + Pr(4) + Pr(5)] = 2=3 and Nfm(F Y )=w(b)[pr(2) + Pr(3) + Pr(4) + Pr(5)] + Pr(6) = 1=3. Nfm(F Z )=0 Let G = ff X F Y F Z g,and(g) be the "fuzzy -algebra" generated by F X, F Y and F Z.Ifwecontinue in this mode, we would be able to make (U (G) Nfm) a probability space. However, this is another topic, we will not do so here. It is developed in other papers [16, 17]. For now we will just dene the bpa's. We setm(x) =Nfm(F X ), m(y )= Nfm(F Y ), and m(z) =Nfm(F Z ) and the rest of bpa's to be 0. The only focal elements are X and Y. Plainly, we use the new measure of "disjoint fuzzy sets" as bpa. In other words, we are using mutually "disjoint" bpa's. Let us compute some examples. Let A = X\Y, then I[A] =.LetB = f g, then I[B] =Y [ Z. So we can compute the Bel by "disjoint bpa's," namely, Bel(A) =m( ) =0 Bel(B) =m(y )+m(z) =1=3 It is clear from this example that the bpa of X and Y consists of weighted measures contributed from individual points. Those points in the intersection,

15 Rough Sets and Belief Functions 15 namely, 2,3,4,5 contribute weighted measures to each focal elements, while "nonintersection" points contributed full measures (weight = 1). So we have accomplished what we have set out for. Namely, the points in the intersection contributed "disjointedly." In the next section, we proceed from granular computing point of view, which is more direct and straightforward. 6.4 From Granular Computing From the arguments of Section 6.3, we hope we have convinced readers and ourselves the following fact: Even though a family of focal elements may not be mutually disjoint, bpa's are a set of mutually disjoint measures. This fact "legalizes" the computation of belief functions. In extended rough set representations, we represent the granular structure of a binary neighborhood system (a binary relation) by syntactically independent words (elementary concepts). The same idea can be applied to general neighborhood system. As an application, we apply it to the family of focal elements. Suppose a neighborhood system (in nite universe), say F = ff1 F2 :: F m g, is given, then a natural bpa m can be dened by m(f j )=Card(F j )= P j=::: Card(F j), So a neighborhood system gives rise to a natural belief function. Conversely, let us assume bpa m is given. Let F = ff1 F2 :: F m g be the family of the focal elements. Then, F forms a (general) neighborhood system: Each F j is a neighborhood of all its point. If a point is not covered by any focal element, its neighborhood is the empty set. So neighborhood systems are the natural settings for belief functions. We can anticipate many interesting interactions between neighborhood systems and belief functions. The interactions may beas fruitful as that of rough set theory. Our aim in this section is to generalize the proposition in Section 6.2. Assume we are given an m, and hence an F. Let C = fc1 C2 :: C m g, where C j = NAME(F j ). Naming these focal elements is very important. Their names should reect the fact that each measure, m(f j ), is mutually disjoint. C is called the concept space of granular structure [13]. Let us dene Pr(C j ) = m(f j ) and extend it additively to all subsets of C. Then, (C 2 C Pr) forms a probability space. Note that F is not a partition, (U (F ) Pr) can not form a probability space. In rough set theory, we consider the lower and upper approximation by elementary sets. In binary relations, we consider the approximation by elementary neighborhoods, namely, the interior I[X] and the closure C[X]. In this section, we will consider the approximation by concepts, that is, the names of neighborhoods. Let X U. Write C A = fc j : C j = NAME(F j ) and F j Ag C A = fc j : C j = NAME(F j ) and F j \ A 6= g

16 16 T.Y. Lin C A and C A are called lower and upper concept approximation respectively. They are collections of words, not subsets of U. Pr(C A)= P F j A Pr(C j) Pr(C A)= P F j \A6= Pr(C j) Now, it is easy to see the following generalization of the proposition in Section 6.2. Proposition. Bel(A) =Pr(C A), and Pl(A) =Pr(C A). Example A granular computing example Let the universe be U = f g. The focal elements and their bpa are: X = f g, andy = f g m(x) =2=3, m(y )=1=3, and other bpa's are 0. Let C = fc1 C2g, where NAME(X)=C1 and NAME(Y )=C2.We dene Pr(C1) = m(x) =2=3 andpr(c2) =m(y )=1=3. Extend Pr additively to 2 C,wehave a probability space (C 2 C Pr). Let A = X \Y, then C A =. LetB = f g,thenC B = fc1 C2g. Compute Bel and Pr with respective denitions and rules, we nd the following relatons, Bel(A) =Pr(C A)=0 Bel(B) =Pr(C B)=1=3+2=3 This veries the proposition. Note that C B and I[B] are dierent. Former is a setofnames,fc1 C2g, it is a subset of the concept space C. Latter is a subset, f g, of the data space U. 7 Conclusions Two of the most signicant results in rough set theory are the representations of equivalence relations and processing of information table (a theory of knowledge). In this paper, we successfully extend the representations and processing to binary relations (or geometrically binary neighborhood systems). We applied the idea to belief functions (in a more general setting than that of binary relations)and gained some new insight to the theory of evidence. These new representations are called extended information tables. We should stress that in the extended representations, words (elementary concepts) are semantically may be

17 Rough Sets and Belief Functions 17 dependent. This is in a strong contrast to rough set theory. In rough sets, elementary concepts represent elementary sets and vise versa. All elementary sets are disjoints, so the semantics of elementary concepts are independent. The processing of words is equivalent to processing of equivalence classes so naming the elementary sets is not critical. On the other hand, in the extended information tables, words may not be independent. The constraints among these words have to be properly dealt with. So naming the elementary neighborhoods is very important. Ideally, these names should reect the semantic constraintsin linguistic forms so that the processing (with human interactions) of these names can be accomplished without referring back to those elementary neighborhoods see discussions on relational databases below. Fuzzy logic handles such constraints beautifully this will be reported in the next paper [14]. In some database applications, such constraints pose no problems, because applications are looking at high level information [13]. However in the computation of belief measures, we do have to observe the constraints of words. Finally, we would like to comment on"word processing" of relational databases and rough set theory. Aswehave mentioned many times, information tables and relations in relational databases are closely related. However, they are dierent in their respective"word processing." In rough set theory, all the word processing is referred back to partitions, for examples, attribute dependencies are checked via partitions. In databases, functional dependencies are determined by the semantics of attribute names (i.e.,intension in database terminology). It is interesting to observe that the "data" in databases are "words" in rough set theory. Some of the "data processing" in databases mayinfactbe"word processing" of extended information tables we will report it in the near future. References 1. S. Bairamian, Goal Search in Relational Databases, Thesis, California State University at Northridge, T. Back, Evolutionary Algorithm in Theory and Practice, Oxford University Press, W. Chu, Neighborhood and associative query answering, Journal of Intelligent Information Systems, 1, , R. Duda, and P. Hart, \Pattern Classication and Scene Analysis,"Wiley- Interscience, K. Engesser, Some connections between topological and Modal Logic, Mathematical Logic Quarterly, 41, 49-64, Grzyma la-busse, J. W.: Rough Sets. Advances in Imaging and Electrons Physics 94 (1995). 7. P. Halmos, Measure Theory, Van Nostrand, Kretowski, M., Stepaniuk, J.: Selection of objects and attributes a tolerance rough set approach. Proc. of the Ninth International Symposium on Methodologies for Intelligent Systems. June 1-13, Zakopane, Poland see also: ICS Research Report 54/95, Warsaw University of Technology (1996)

18 18 T.Y. Lin 9. T. Y. Lin, Neighborhood Systems and Relational Database. In: Proceedings of 1988 ACM Sixteen Annual Computer Science Conference, February 23-25, 1988, T. Y. Lin, Neighborhood Systems and Approximation in Database and Knowledge Base Systems, Proceedings of the Fourth International Symposium on Methodologies of Intelligent Systems, Poster Session, October 12-15, pp , T. Y. Lin, Topological and Fuzzy Rough Sets. In: Decision Support by Experience - Application of the Rough Sets Theory, R. Slowinski (ed.), Kluwer Academic Publishers, , T. Y. Lin, Neighborhood Systems -A Qualitative Theory for Fuzzy and Rough Sets. In: Advances in Machine Intelligence and Soft Computing, Volume IV. Ed. Paul Wang, , Also in Proceedings of Second Annual Joint Conference on Information Science, Wrightsville Beach, North Carolina, Sept. 28-Oct. 1, 1995, , T. Y. Lin, Granular Computing of Binary relations I:Data Mining and Neighbrohood Systems. In:Rough Sets and Knowledge Discovery Polkowski and Skowron (Editors), Springer-Verlagm (to appear). 14. T. Y. Lin, Granular Computing: Fuzzy Logic and Rough Sets. In Computing with words information/intelligent systems L.A. Zadeh and J. Kacprzyk (Editors), Springer-Verlag, (to appear) 15. T. Y. Lin, and M. Hadjimichaelm M., Non-classicatory Generalization in Data Mining. In: Proceedings of The Fourth Workshop on Rough Sets, Fuzzy Sets and Machine Discovery, Tokyo, Japan, November 8-10, , T. Y. Lin, Sets with Partial Membership: A rough set view of fuzzy sets, San Jose State University, Pre-print, T. Y. Lin, Additive Fuzzy Measure and Belief Functions: A Granular View of Fuzzy Sets, San Jose State University, Pre-print, T. Y. Lin and C. J. Liau, Probabilstic Multivalued Random Variables: Belief Functions and Granular Computing. In Proceedings of 5th European Congress on Intelligent Techniques and Soft Computing, Aachen, Germany, September 8-12, 1997, T. Y. Lin, and J. C., Liau, Belief Functions Based on Probabilistic Multivalued Random Variables, in: The Proceedings of Third Joint Conference of Information Sciences, Research Triangle Park, North Carolina, March 1-5, , T. Y. Lin and Q. Liu First Order Rough Logic I: Approximate Reasoning via Rough Sets, Fundamenta Informaticae. Volume 27, Number 2,3, 1996, T. Y. Lin, and Y. Y. Yao, Mining Soft Rules Using Rough Sets and Neighborhoods. In: Symposium on Modeling, Analysis and Simulation, CESA'96 IMACS Multiconference (Computational Engineering in Systems Applications), Lille, France, 1996, Vol. 2 of 2, , Z. Pawlak, Rough Probability, Bull. Pol. Acd. Sci., Math 32, , Z. Pawlak, Rough sets. Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Polkowski, L., Skowron, A., and Zytkow, J., (1995), Tolerance based rough sets. In: T.Y. Lin and A. Wildberger (eds.), Soft Computing: Rough Sets, Fuzzy Logic Neural Networks, Uncertainty Management, Knowledge Discovery, Simulation Councils, Inc. San Diego CA, 55{ G. Shafer, A Mathematical theory of Evidence, Princeton University, W. Sierpenski and C. Krieger, General Topology, University of Torranto Press 1956.

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