Granular Computing: Examples, Intuitions and Modeling

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1 Granular Computing: Examples, Intuitions and Modeling Tsau Young (T. Y.) Lin, Member; IEEE, Abstract- The notion of granular computing is examined. Obvious examples, such as fuzzy numbers, infinitesimal number and access control model, (pre-)topological spaces are examined. A general models are proposed; Localized multi-level granulation can be modeled by generalized topological spaces, called neighborhood systems. For most general granulation are modeled by Tarski type relational structures. Index Terms- binary relation, granular computing, neighborhood system, topology. I. INTRODUCTION Granulate and Conquer seems like a natural problemsolving methodology deeply rooted in human thinking. Human body has been granulated into head, neck, and so forth; geographic features into mountains, planes, and others. Many daily "things" have been routinely granulated into sub"things." The notion is ancient, and intrinsically fuzzy, vague, imprecise; It is difficult to deal with. So almost equally ancient, mathematicians idealized it into the notion of partitions, and developed it into a fundamental problem-solving methodology; it has played major roles throughout the entire history of mathematics. Nevertheless, the notion of partitions, which absolutely does not permit any overlapping among its granules, seems to be too restrictive for real world problems. Even in natural science, classification does permit small degree of overlapping; there are beings that are both proper subjects of zoology and botany. A more general consideration is needed. Based on Zadeh's grand project on granular mathematics, during his sabbatical leave (1996/1997) at Berkeley, Lin focused on a subset of granular mathematics, which he called granular computing [?]. To stimulate research on granular computing, a special interest group, with T. Y. Lin as its Chair, was formed within BISC (Berkeley Initiative in Soft Computing). Since then, granular computing has evolved into an active research area, generating many articles, books and presentations at conferences, workshops and special sessions. Some Historical Notes. Though the label, granular computing is relatively recent, the notion of granulation has in fact been appeared, under different names, in many related fields. However, for more systematic development of GrC in computer science that are non-partition theory has two. The first ones is actually buried in the design of fuzzy control systems [34], [35],[21] [22], [23]. However, the explicit mentioned the term granularity and discussing its Tsau Young Lin is with Department of Computer Science, San Jose State University /05/$20.00 C2005 IEEE concept is in the article [?]; its newest version is in [?]. The second group is related to databases. The first database papers that have considered non-partition theory probably belong to S. Ginsburg [3]. They considered the binary relation of order. Totally, from different directions, we convert Motro's numerical metric [25] into qualitative metric, to enhance approximate retrieval in database, [9], and case based reasoning [?]; the idea was followed by [2]. We have called such a mathematical system A neighborhood system (NS). NS is actually a set of binary relations, and is a overlapped with a subject. Called Frechet(V) space in the book [28]. In 1992, we related the idea to rough set theory in[12]. II. EXAMPLES A. Fuzzy Numbers From [32], a fuzzy real number M is a convex normalized fuzzy set M, whose membership function is JM, of real line R such that 1) It exists exactly one point po with pm (PO) = 1 (po is called mean value of M) 2) pam(po) = 1 is piecewise continuous. In other words, the set of fuzzy real numbers, RF is the set of membership functions IaM whose mean value po varies through the classical real line. So we can imagine that the set of fuzzy real numbers is constructed by attaching to each real number r a fuzzy set pm at its mean value r = po. So. The set of fuzzy real numbers, may also be referred to as the fuzzy real line, is constructed by inserting a fuzzy set to each classical real number. Formally, we proceed as follows: Definition 1: Let MF(R)be the set of all membership functions on the set of classical real numbers R. Then the map R ) MF(R): po - AM, is a fuzzy basic granulation, denoted by RF B. Infinitesimal Numbers From [31], an infinitesimal is a number that is greater in absolute value than zero yet smaller than any positive real number. A number x 74 0 is an infinitesimal iff every sum -x x- of finitely many terms is less than 1, no matter how large the finite number of terms. In standard analysis, infinitesimal is only a notional quantity, and there exists no infinitesimal real number. The first mathematician to make use of infinitesimals was Archimedes, although he did not believe in their existence. When Newton and Leibniz developed the calculus, they made use of infinitesimals. However, the use 40

2 of infinitesimals was attacked as incorrect by Bishop Berkeley. The fundamental problem is that infinitesimal is often treated as non-zero (division, e.g, dy/dx is permissible), but often discarded later as if it were zero (standard number + infinitesimal = standard number, e.g., x + dx = x)). It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit, which obviates the need to use infinitesimals. Nevertheless, the use of infinitesimals continues to be convenient for simplifying notation and calculation. Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson. In this theory, the above computation can be justified with a minor modification: we have to talk about the standard part of the difference quotient (e.g., dy/dx), and the standard part of e.g., x + dx is x. In other words, to each standard number we insert. The set of non standard real numbers is constructed by inserting the set of infinitesimals to each classical real number. As formal theory is too lengthy, we will skip here. C. Access Control We have given two examples of localized notion, namely, to each object, we insert a crisp or fuzzy subset into the universe. Now we will investigate, instead of inserting a subsets, but a new constraint. Namely, to each object, there is a constraint associate to it. For example, to each object, we are allowed or disallowed to access some nearby objects, the so called access control model in computer security. Abstractly, for each object p E V, we have a granule B(p) C U of objects which are in conflicts, or more plainly a list (granule) of enemies. Such a view leads to Chinese wall security policy model [20], [10], [1]; there are other applications [30]. Note that the association of each object a list of object is mathematically a binary relation. Theorem 1: Chinese Wall Security Policy Theorem. Let V = U and B be conflict of interests. If B is anti-reflexive, symmetric and anti-transitive, then the complement of B is an equivalence relation, so no information will flow to enemy hands. D. Sensor Networks The previous three examples, are static granulation, now we will have a dynamic one: Suppose we are in an environment that has many sensor scattered around, then A moving target will trigger a granule of background sensors start to Communicate or sense the moving target. As the target move, a different granules of background will be activated. This will be an interesting new application area of granular computing. E. Tangent Spaces Instead of real line, we may consider infinitesimal on the high dimension spaces. In this case, the infinitesimal forms a tangent plane locally. In fact, we may consider a subset of Euclidean space. For example, cons?'der a sphere in the Euclidean 3-space, we have a tangent plane at each point. These tangent planes are generated by those infinitesimals. This idea has been generalized to fibre bundle, or fibre spaces [29]. F General Topology Roughly speaking the examination of infinitesimal may be viewed as an analysis of the notion of near or neighborhoods. Its abstraction leads to a formal subject, called general topology. The notion of near is rather imprecise. Let us examine the following two examples. Example 1 Is Santa Monica "near" Los Angels?. Answers could vary. For local residents, answers are often "yes." For visitors who have no cars, answers may be "no." Example 2 Is 1.73 "near"?. Again answers vary. Intrinsically "near" involves some subjective judgments. One might wonder whether there is a scientific theory for such subjective judgments? Mathematics has offered a nice solution. It simply includes the contexts into the formalism: Given the radius of an acceptable error, say, < 1/100, is 1.73 "near"? Similarly, if a neighborhood system has been assigned to each city in Los Angeles area, then we have a definite answer for Example 1. A proper formulation for such a question is: 1) Given a neighborhood system(ns), is Santa Monica "near" Los Angels? 2) Given the radius of an acceptable error ( =1/100(NS), is 1.73 "near"? 41 The formal subject of studying the notion of near is called General topology, which can be formulated in two forms 1) Localized Version: V is called a topological space, if for each point p E V, we assign a family TNS(p) of subsets, called neighborhoods, that meets certain "topological axioms;" for this paper the precise axioms are not important. The collection TNS(V)={TNS(p)l p E V } is called the topological neighborhood system of V; we may simply use TNS, if the context is clear. 2) Global Version: V is called a topological space, if a family of subsets, called open sets, that are given and satisfies the topological axioms. Again the precise axioms are not important here. The family of open set will be denoted by r(v), or simply, F is called the topology of V. G. Combined Notion - Neighborhood Systems The structure of general topology is too rich for computer science applications, since such a rich structure produces many pathological phenomena when the universe is a finite set; some relaxation is needed. The sections of II-A, II-B and ll-d can be viewed as an instance of topology. Namely, one looks at one neighborhood, e.g., a fuzzy set, a set of infinitesimals, asset

3 of enemies. Algebraically such a single view at each object s a binary relation. So a naturally extended the notion, that combines the binary relation and TNS, has been called a neighborhood system, we often use LNS, as to distinguish it from TNS, to denote it; we may use NS to denote the combined cases [9]. Our aim is to formalize (part of) granulation in terms of LNS and TNS. So a neighborhood has been termed a granule too. Definition 2: Let V and U be two universes (classical sets). For each object p E V, we associate a family of crisp/fuzzy subsets, denoted by NS(p)C P(U), that is, we have a map B: V -* 2P(U) : p ~-4NS(p), where P(U) means all crisp/fuzzy subsets on U, and 2P(U) means the power set of all crisp/fuzzy subsets. NS(p) is called a crisp/fuzzy neighborhood system at p. NS(U)={NS(p) p E V } is a neighborhood system of V. In the case NS(p)={B(p)} is a crisp/fuzzy singleton, we define Definition 3: The map B: V - P(U), binary granulation (BG) and the collection {B(p) p E V} a crisp/fuzzy binary neighborhood system (BNS). We should observe that B(p) is a neighborhood (granule) of p implies p E B(p), but it does not imply that q E B(p) implies B(p) is a neighborhood of q. It is clear that given a map B gives rise to a crisp/fuzzy binary relation BR: BR={(p,x) x E B(p) andp E V}. In both crisp/fuzzy cases, BR, BNS, and BG will be treated as synonyms. III. INTUITION AND FORMALIZATION Combine these examples, we may reach a general notion of granulation. In addition, we will carefully analyze Zadeh's assertion. A. Localized and Global Granulation The notion of LNS can be viwed as most general notion of (multi-level) localized granulation of V. The r can be viewed as a "typical" notion of multi-level global granulation on V. Basically, if a family of granules is attached to some points, we will regard them as localized multi-level granulation. If the family is single ton, then it is single level localized granulation; we name the last case a basic granulation, or binary granulation(because it can be derived from a binary relation). If such a family of granules is not attached (or attached to every point of every granule), we call it global multi-level granulation. For this case, the only single level granulation is partition. As the consequence of this observation, a family of closed sets, open sets, measurable sets, random sets, partitions and etc are all global granulations. The localized multi-level granulation were call neighborhood systems [9], [12]; these notions were developed prior to the label of granular computing was proposed by this writer [37]. 42 B. Zadeh's View and Relational Structure According to Lotfi Zadeh [36]. "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.". The phrase "drawn together by indistinguishability, similarity or functionality," in general, can be expressed mathematically by relations. If the group of drawn together consists of n objects, then the relation is n-ary; we may refer to such structure as n-ary granulation. In general, for every n, there may have several, even infinitely many, n-ary relations. In granular computing, n can be any cardinal number. In relational structure of the model theory, they are all finite; first order logic do not use predicates of non-finite places. In real life, we seldom beyond initial few n, one of the most useful mathematical structure is topological space, which is the case of n = 2. Definition 4: By a granulation on V we mean a family of n-ary relations, for different n, including infinite, are given on V. This is basically an extension of Tarski's relational structure. We should note that Zadeh was not interested in such a mathematical structure per se; he is interested in the case that semantic considerations, such as indisitnguishability, similarity and functionality, may be led to such a structure. In many daily considerations, n is very often equal to 2. IV. GRANULAR COMPUTING AND KNOWLEDGE PROCESSING Computing the granules can be regarded as knowledge processing. Granules can be regarded as the basic units of knowledge. So the granular representations, in which each granule is represented by a human perceived notion, are vital important in information processing; the basic theory was developed in [13], [14], [15], [18], [19]. One of the simplest knowledge processing, which is developed by rough set theory [8], [26], is approximation, that is an unknown concept (a subset of the universe) can be approximated by the known granules. A. Approximations In this phase, granules are regarded as known knowledge, So the effort is to approximated a new concept by granules. There are many ways to achieve this. Some basic notions are limit points, derived sets [28]), closure, interior points [12], upper and lower approximations [26]. Note that lower and upper approximation do not equal to interior and closure in non-partition cases. a subset of U. 1) I[X] = {p : there is an N(p) that is a subset of X} = Interior in [12], we say it is the largest open set is inaccurate. 2) C[X] = {p : For every N(p) such that N(P) has nonempty intersection with X} = Closure; in [12], we say it is the smallest closed set is a wrong statement. 3) L[X] = U{B(p) : for all B(p) that is a subset of X} = Lower approximation; this is not a very meaningful notion in NS, but is useful in measure theory

4 4) H[X] = U{B(p): For every N(p) such that N(P) has non-empty intersection with X} = Upper approximation; this is not a very meaningful notion in NS, but is useful in measure theory 5) Upper and lower approximation do not equal to interior and closure in non-partition cases. 6) Let (U, A) be an approximation space (A is a partition). BNS is a clopen topology. 7) An object p is a limit point of a set X, if every neighborhoods of p contains a point of X other than p. The set of all limit points of X is call derived set D[X]. 8) Note that C[X] = X U D[X] may not be closed. 9) Some authors (e.g.[28]) define the closure as X together with repeated (transfinite) derived set, that is, C[X] = X U D(... (D[X])...) For such a closure it is a closed set. V. CONCLUSIONS Granulate and conquer is a natural problem solving strategy since ancient time. Partition, the idealized form, has played a central role in the history of mathematics, recently in the rough set theory. However, in this paper, our interest on non-partition theory. Granulation is a powerful notions; The general form is an intrinsic part of fuzzy control [35] [21], [22], [23]. When we restricted to the binary case, n=2, it can be viewed as a generalization of classical topology and binary relations. Hence is a generalization of rough set theory. It is very useful in approximate retrieval [9], [11], [2], [27], [24], and data mining mining [17], [16] [6], and other areas [4], [7]. A very cute application of "extremely non-partition" theory to security is explained; it explore the possibilities of the control of Trojan horses. REFERENCES [1] David D. C. Brewer and Michael J. Nash: "The Chinese Wall Security Policy" IEEE Symposium on Security and Privacy, Oakland, May, 1988, pp , [2] W. Chu and Q. Chen Neighborhood and associative query answering, Journal of Intelligent Information Systems, 1, , [3] Seymour Ginsburg, Richard Hull: Order Dependency in the Relational Model. Theor. Comput. 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