Sets with Partial Memberships A Rough Set View of Fuzzy Sets T. Y. Lin Department of Mathematics and Computer Science San Jose State University San Jose, California 95192-0103 E-mail: tylin@cs.sjsu.edu Abstract Fuzzy set theory on a universe of granules (crisp sets) are considered. In this setting, the grade of a membership can be interpreted in the most elementary fashion. A grade of 1/2 membership means exactly one half of the granule (in a suitable measure) belongs to the fuzzy set. Based on such views, fuzzy set theoretical operations can be very naturally induced from the classical set theoretical operations; they are called R-operations. Some traditional operations, such as Max and Min, are R- operations of specific classes of fuzzy sets. The granular view of fuzzy sets provides a new and healthy insight into how fuzzy set theory might be structured. 1. Introduction Fuzzy sets are defined by membership functions [1]. There are various ways to interpret the grades of memberships. In this paper, we take the most elementary interpretation, when the universe is a crisp set of granules (crisp sets). In such a setting, a partial membership does have its natural meaning. Let a fuzzy set be defined on the universe U of granules. Let F be its membership function. Assume F (x) = 1/2 for a granule x. In this case the grade 1/2 means exactly one half of the granule belong to the fuzzy set. Based on such interpretations, the usual fuzzy set theoretical operations can be defined very naturally in terms of classical set operations. It is clear that such operation is not truth functional. In other words, the membership function of the intersection of two fuzzy sets is not a function of the membership functions of these two fuzzy sets. Similar remarks hold for the union. So the results are not compatible usual s- and t-norms [2]; s- and t-norms are based on the assumption of truth functional. In this paper, we emphasize on presenting the idea. So we focus on simple cases, namely, the universe is finite and the measure is the counting measure (cardinal numbers) so we can view the idea from rough set theory. More elaborate theory and based on different philosophy will be presented in other papers. We hope this idea can give fuzzy set theory a new foundations. 2. Partial Memberships-Intuitive Considerations Let us examine some "common sense" example. If Mr. Smith works 20 hours a week for Company, we often refer him as a half time employee of Company. Can we express his membership mathematically? The answer is "yes": Let F be a membership function, then Mr. Smith's grade of membership is F (Smith)=20/40. Our goal is to develop a fuzzy theory of this type. Assume further Mr. Smith also works for Company Y and Z for 10 hours each. By the same arguments, his membership grade will be 1/4 for both companies. What would be the grade of his total employment? Let us assume he works 40 hours a week. (1) If companies are totally unrelated, then his total employment would be 20/40 + 10/40 + 10/40. (2) If both Y and Z are subsidiary of, then his total employment could be still 20/40. (Y's and Z's records are merely items record of ) We can interpret, Y and Z as fuzzy sets, and their membership grades for Mr. Smith are 20/40, 10/40 and ---------------- This research is partially supported by Electric Power Research Institute, Palo Alto, California, San Jose State University, Amdhal Corporation, NASA Grant NCC2-275, ONR Grant N00014-96-1-0556, LLNL Grant 442427-26449, ARO Grant DAAH04-961-0341, and BISC Program of UC Berkeley.
10/40 respectively. In other words, we have a granule 40 hours, and this granule is subdivided into three subgranules: 20 hours, 10 hours, and 10 hours. The grade of memberships are the ratio of sub-granule and whole granule. 3. Upper and Base Universe of Discourses Our goal is as follows: Let F be a membership function on C, a set of disjoint granules. To each granule of C, we want to associate a sub-granule such that the grade of F at a granule is the ratio of two measures, the measures of the granule and its sub-granule. For reasons that will be clear, we call C the base universe of discourse. Let U, called the upper universe, be the disjoint union of these granules. These granules form a partition on U. Let P be the partition or the corresponding equivalence relation. Each equivalence class plays two roles: First it is a subset of U, second it is an element of U/P. To make the exposition simpler and clearer, we will name these equivalence classes. Let G 1, G 2, G 3,, G n denote these equivalence classes (granules) in U, and c 1, c 2, c 3,, c n be their respective names, that is, c j =NAME(G j ). They are elements of the base universe C, which is isomorphic to the quotient set of U/P. The goal of this paper is to build a fuzzy set theory on C using U as a representation space. So that we may understand the nature of fuzzy sets better. We will illustrate the idea in Figure 1. In the following figure, the large rectangle denotes the upper universe U. Small rectangles G j denote equivalence classes. Black dots denote c j =NAME(G j )'s. Each vertical arrow represents the projection that maps each G j into its name c j. The family of all small rectangles is the quotient set U/P that is isomorphic to C, the set of black dots. c 1 c 2 c 3 c 4 c 5 c n- c n 1 Figure 1 Such a structure is a special structure that we introduced in [3], called a granular structure. Definition A granular structure consists of 4-tuple (V, U, B, C) where V is called the object space, U is the data space (V and U could be the same set), B is a crisp/fuzzy neighborhood system, and C is the concept space which consists of the names of the fundamental neighborhoods of B. If B is a binary neighborhood system (binary relation), then the 4-tuple (V, U, B, C) is called a binary granular structure. In the present case, V=U and B is a partition (equivalence relation). Each equivalence class is the neighborhood of each of its points in the class. We will abbreviate the 4- tuple as 3-tuple (U, P, C), where P stands for partition and C is the name of all the equivalence classes. Note there is a natural projection U U/P C; so we will, by abuse of notation, denote the projection, the partition, or the equivalence relation by P. 4. Realizations of Membership Functions Our next topic is to show how one can construct a crisp set representation for each membership function on C. We will illustrate the idea by examples. For simplicity, let us assume n = 9 and all equivalence classes consist of 5 elements. Suppose F be a membership function defined as follows: F(c 1 ) = 1/5, F(c 2 ) = 2/5, F(c 3 ) = 3/5, F(c 4 ) = 4/5, F(c 5 ) = 5/5, F(c 6 ) = 4/5, F(c 7 ) = 3/5, F(c 8 ) = 2/5, F(c 9 ) = 1/5. G 1 G 2 G 3 G 4 G 5 G n fl fl fl fl fl fl fl Our goal is to construct a subset of U such that F can be expressed in terms of and G j as follows: Let denote the cardinal number. F(c 1 ) = G 1 / G 1, F(c 2 ) = G 2 / G 2, F(c 3 ) = G 3 / G 3, F(c 4 ) = G 4 / G 4, F(c 5 ) = G 5 / G 5, F(c 6 ) = G 6 / G 6, F(c 7 ) = G 7 / G 7, F(c 8 ) = G 8 / G 8,
F(c 9 ) = G 9 / G 9. In Figure 2, each square represents one element of U. All the squares in the same vertical column belong to the same equivalence class G j. Let j denote the subset of G j that consists of j squares located below the square j. Note that j = j. 5 4 6 3 7 2 8 1 9 fl fl fl fl fl fl fl fl fl Figure 2 From the figure, it is easy to verify that j / G j = j/5 =F j (c j ), if j 5 j / G j = (j-5) /5 =F j (c j ), if j > 5 Let = 9 j=1 j. Then is the desired subset. will be called a realization of F. Of course, the realization is not unique. Theorem. Let C be a crisp set, called the base universe. Let F be a rational valued membership function defined on C. Then there is a crisp set U, called the upper universe, such that (1) there is a partition P ={G j : j =1,2,..n} on U, namely, G j are disjoint and U = n j=1 G j, such that the quotient set U/P is isomorphic to C, and (2) there is a realization U such that F(c j )= G j / G j. Once we found the upper universe U, we may treat each element of C as a name of an equivalence class U. We should also assert that the realization may not be unique. The restriction, rational valued, that we impose in last theorem can be removed. The restriction was there because we use the counting measures. If we are willing to use measure theory, such a restriction disappears. Theorem. Let C a base universe, at most countably infinite. Let F be a membership function defined on C. Then there is a measure space (U, µ) and a measurable partition P (i.e., each equivalence class is measurable) such that (1) P: U C is the map that induces the isomorphism U/P C (P is the projection and the partition) (2) There is a measurable set U such that F(c) = µ ( P -1 (c))/ µ(p -1 (c)) The easiest U is the direct product of the unit interval and C. Since C is at most countable, we can use the counting measure. The desired measure on U is the product of counting measure on C and the Lebesque measure on the unit interval. The term upper universe shall mean the triple (U, µ, P); we use U to denote the triple. Definition: 1. The constructed in the theorem is called a realization of the membership function F. 2. The pair (, F) is called granular fuzzy set and F granular membership function. The term fuzzy set in this paper is always referred to the pair. 3. If µ(p -1 (c))=1 for all c C, then (, F) is called a normalized fuzzy set. For simplicity, we assume all fuzzy sets are normalized through out the rest of the paper. Note that a granular membership function induced a membership function on U R(u)=F(c), where P(u)=c. Such R is called rough membership function in [4]. 5. Fuzzy Set Theoretical Operations Whenever a fuzzy set with membership function is given, we will assume that we have fixed a realization.
Definition of R-operations The set operations for such fuzzy sets are called R- operations (Realization operations). Let (, F ) and (Y, F Y ) be two fuzzy sets. Then (1) the union is defined by (, F ) (Y, F Y ) = ( Y, F ( Y) ) F ( Y) (c)=µ[( Y) P -1 (c)] (2) the intersection is defined by (, F ) (Y, F Y ) = ( Y, F ( Y) ) F ( Y) (c)=µ [( Y) P -1 (c)] From definitions, it is clear that the membership functions F ( Y), F ( Y) of R-operations are very different from the usual s- and t-norms. However, the context free fuzzy sets that we introduced in [5,6] are compatible with these operations. 5. Max and Min Operations From Figure 3, it is easy to see that F ( Y) = F + F Y F ( Y) = 0. It is clear that the intersection can not be computed from its components. In other words, F ( Y) or F ( Y) is not a function of F, F Y ; Such a property is commonly referred to as "R-operations are not truth functional." Example 2 1.1. Let be a realization of F indicated in Figure 4. Each j denotes all the boxes below it. 1.2. Similarly Y be a realization of F Y. As above each Y j denotes all the boxes below it. 4 5 2 3 8 Example 1 1.1. Let be a realization of F indicated in Figure 3. Each j denotes all the boxes below it. 1.2. Let Y be a realization of F Y. Each Y j denotes all the boxes above it. 1 6 7 Y 9 9 Y 1 Y 4 Y 5 Y 2 Y 3 Y 8 Y 6 Y 7 Figure 4 1 6 2 3 4 5 7 8 9 Figure 3 Y 1 Y 6 Y 2 Y 3 Y 4 Y 5 Y 7 Y 8 Y 9
Figure 5 From Figure 4 and 5, it is easy to get F ( Y) = Max (F, F Y ) F ( Y) = Min (F, F Y ) For this example, the granular membership functions F ( Y), and F ( Y) are the same as usual Max and Min operations. The reason is that both and Y take their realizations "canonically" (see remark). In this particular example, both and Y take the squares from bottom up. However, in Example 1, and Y are taken from different areas. These two examples clearly show that the Max and Min only work for certain specific realizations. Remark: The "canonical" way is of course ad hoc: There are many ways. One possible way may go as follows: Well order an equivalence class (well ordering principle) and fixed that order. So the equivalence class may looks like, x 1, x 2,, x j,, x n. To choose a subset of size j, one always starts from selecting the first element and stop some j-th element. In other words, all subsets look like x 1.. x 1, x 2,, x j. x 1, x 2,, x j,, x n This shows R-operations represent Max and Min by making ad hoc choice; they are not compatible. In [5,6], we developed the context free fuzzy sets; their operations are called F-operations. It turns out that F- operations are compatible with R-operations. We will give a summary in next section. 5. CONTET FREE FUZZY SETS Context free fuzzy sets were developed earlier [5,6] from the following consideration: Fuzzy sets mathematically are merely real-valued functions. So one should not force the set operations into fuzzy sets just by mere analogy to classical set theory. By taking the most conservative stand, no operations are used to create a new membership function from two given ones. We simply "invent" a "new" type of fuzzy sets. In this new theory, no new membership function is created from two membership functions; we merely "re-define" a "new fuzzy set" that is characterized by two membership functions. In other words, fuzzy set is no longer defined by a unique membership function, it is in fact, defined by a finite set of membership functions. Two membership functions may form a new membership function, only if their respective realizations become available. Definition 5.1 1. A primitive fuzzy set is defined by and only by a membership function F: U [0. 1] 2. A multi- fuzzy set is defined by and only by a set of membership functions, F = {F 1, F 2,..., F n }. Intuitively, a primitive fuzzy set is an abstract object that carries information provided by one single membership function. A multi-fuzzy set is an abstract object that carries information provided by a finite collection of membership functions. Note that the properties of a primitive fuzzy set are solely represented by its membership function [2, pp.12]. Similarly those properties of a multi-fuzzy set are solely represented by the collection of membership functions. Multi- fuzzy sets can be represented high dimensional fuzzy sest. Let M m be the m-dimensional Cartesian product of [0, 1]; we may refer M m as little cube. Proposition 5.2 A multi-fuzzy set can be characterized by the map F: U M m defined by u (F 1 (u), F 2 (u),..., F m (u)), where each F i (u) is a membership function. The map is called a m-dimensional membership function, and is denoted by F 1 Λ F 2 Λ... Λ F n. It can be organized into a table format; each membership function represents a column. U F 1 F 2 F 3... U 1.1.12.13... U 2.2.13.12.................. U n.2.15.09... We will define the INTERSECTION simply by taking the union of two collections of membership functions. Definition 5.3. Let F and FY be two multi-fuzzy sets.
Then the INTERSECTION is defined by F Λ FY = F 1 Λ... Λ F m Λ FY 1 Λ... Λ FY n It is an abstract object that carries information provided by "product" of two collections F Λ FY; using database language, the intersection is the join of relations F and FY. In terms of little cubes, F Λ FY is a (m+n) dimensional membership function. The operation Λ is essentially a set theoretical union of membership functions, so algebraically it obeys the usual set theoretical rules, such as (1) F Λ F = F (Idempotent), (2) F Λ FY = FYΛ F (Commutative), and (3) F Λ FΦ = FΦ Λ F = F, where FΦ is the multi-fuzzy set of the empty collection. For the UNION of multi-fuzzy sets, we define it by or. Definition 5.4. Let F and FY be two multi-fuzzy sets. Then the UNION is defined by the expression F + FY = F 1 Λ... Λ F n + FY 1 Λ... Λ FY n It is an abstract object that carries information provided by either F or FY. Using database language, the union is a database consisting of two relations. Now we will define our final objects. Definition 5.4. A Context free fuzzy set is a finite collection of multi-fuzzy sets F, FY,..., FW, denoted by Θ = F + FY+... + FW The empty collection is denoted by. Note that and Θ are uniquely determined by each other. Theorem5.1. The set of context free fuzzy sets are closed under INTERSECTION and UNION. Algebraically, it is a Boolean ring with and FΦ being the zero and identity elements respectively, namely it has "correct" algebraic structure. We hope the meaning of Max-Min-fuzzy-sets is obvious. Theorem 5.2. There is a homomorphim from the algebra of context fuzzy sets to the algebra of granular fuzzy sets (hence to the algebra of Max-Min-fuzzy-sets). This theorem allow us to compute the R-operations along the way of processing fuzzy sets. The results will always be consistent, they are independent of when the realizations are available. 6. Conclusions In this paper, we study a very special types of fuzzy sets. Namely, they are fuzzy sets defined on a universe of "large elements.." More precisely, each member of the universe is a measurable space. Such fuzzy sets allow us to interpret their membership functions in the language of measures of crisp sets. Fuzzy set theoretical operations, base on such interpretations, are defined via classical set operations. We call such operations R- operations. It turns out that these R-operations are incompatible with traditional s- and t-norms. From R- operations point of view, each specific s- and t-norm is operations based on specific assumptions. We illustrate the idea on Max and Min operations. This study indicates that usual s- and t-norms might be inadequate; more research on fundamental set theoretical operations of fuzzy sets needs to be conducted. At the same time, we also please to inform the readers that the F-operations of context free fuzzy sets are compatible with R-operations. This implies that context free direction might be a "right" direction for "universal" operations. References [1] L. A. Zadeh. Fuzzy sets, Information and Control, 8, 338-353, 1965. [2] H. J. Zimmermann, Fuzzy Set Theory --and its Applications, 2nd ed., Kluwer Academic Publisher, 1991. [3] T. Y. Lin. Granular Computing on Binary Relations I: Data Mining and Neighborhood Systems, In: Rough Sets and Knowledge Discovery, Polkowski and Skowron (Eds), Springer-Verlag (to appear). [4] Z.Pawlak and A. Skowron, "Rough Membership Functions." In: R.R. Yager, M.Fedrizzi, and J.Kacprzyk (eds.) Advances in the Dempster- Shafer Theory of Evidence, John Wiley and Sons, New York, 1994, 251-271. [5] T.Y. Lin. Context Free Fuzzy Sets, Proceedings of Second Annual Joint Conference on Information Science, Fourth Annual International Conference on Fuzzy Theory and Technology, Wrightsville Beach, North Carolina, Sept. 28-Oct. 1, 1995, pp. 518-521. [6] T. Y. Lin. A Set Theory for Soft Computing, Proceedings of 1996 IEEE International Conference on Fuzzy Systems, New Orleans, Louisiana, September 8-11, 1996, 1140-1146. Tsau Young (T. Y.) Lin received his Ph.D from Yale University, and now is a Professor at San Jose State
University, also a visiting scholar at BISC, University of California-Berkeley. He is the founding president of international rough set society. He has served as the chairs, co-chairs, and members of program committees in conferences, special sessions and workshops. He is an editor and a member of editorial board in several international journals. His interests include approximation retrievals and reasoning, data mining, data security, data warehouse, fuzzy sets, intelligent control, non classical logic, Petri nets, and rough sets (alphabetical order).