Granular Computing: The Concept of Granulation and Its Formal Theory I

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1 Granular Computing: The Concept of Granulation and Its Formal Theory I Tsau Young (T. Y.) Lin Department of Computer Science, San Jose State University San Jose, California 95192, USA tylin@cs.sjsu.edu 1 Introduction Granulation seems to be a natural methodology deeply rooted in human mind. Daily objects are routinely granulated into sub objects. For example, human body and the surface of earth are often granulated into sub objects, such as head, neck,... and plateaus, hills, plains... respectively. The boundaries of these sub objects are intrinsically fuzzy, vague and imprecise. Flawlessly formalizing such a concept has been difficult. Early mathematicians have idealized/simplified granulation into partitions (=equivalence relations), and have developed the partition into a fundamental concept in mathematics, for example, congruence in Euclidean geometry, quotient structures (groups, rings, etc) in algebra, the concept of a. e. (almost every where) in analysis. 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 appropriate subjects of zoology and botany. So a more general theory, namely, Granular Computing (GrC) is needed: What is Granular Computing (GrC)? It has been a shifting paradigm, since the inception of the idea. Lately, however, the concept seems to have reached its steady state. The goal of this paper is to present this final formal model; of course, nothing can be really final. To trace the intuition of the idea, let us recall how the term was coined. In the academic year , when I arrived at UC-Berkeley for my sabbatical leave, Zadeh suggested granular mathematics (GrM) to be the area of research. To narrow down the scope, I proposed the term granular computing to label the area [42]. Therefore, at the very beginning, GrC is the computable part of the granular mathematics. What is Granular Mathematics (GrM)? In his 1979 paper [40], Zadeh already had implicitly explained his view. In 1997, he outlined his main idea in the seminal paper [43], where he said Basically, TFIG humans its foundation and methodology are mathematical in nature. 1

2 Our views are the same, but I adopted the incremental approach: By mapping Zadeh s intuitive definition [41] to Neighborhood System(NS), I took NS as the GrM, and regarded it as the first GrC model [17], [18], [19]. What is Neighborhood Systems(NS)? Totally from differnt context, namely, approximate retrival, in , I generalized the notion of tolological neighborhood systems (TNS) to that of Neighborhood Systems(NS) by simply dropping the axioms of topology [9], [11]. In this view, each neighborhood is a unit of uncertainty; see Section 5. But in later part of the same year (1989), when I considered the Chinise wall computer security policy model (CWSP), a neighborhood was regarded as a unit of information (known basic knowledge) [10]. In other words, in early GrC period, a granule is a unit of knowledge or lack of knowledge (uncertainty) Since then 9 working models have been built, and it seems that we have reached the steady state. So one of them (8th GrC MNodel) is selected to be the final model. This paper defines the concept of granulation by three approaches: 1. Inducitve Definition: Granulation is defined by a set of examples 2. Zadeh s Informal Definition. 3. The Final Formal Model This final model is a category theory based model (8th GrC model) that fits Zadeh s intuition, and can be specifed to those 9 models and classical examples. The paper is organised as follows: After the introduction, we present a set of classical examples and regard it as an implicit inductive definition of GrC. Then main part, the formal definition of granulation, is presented. In this section, we also include Zadeh s informal definition. Rest of the paper is devoted to reduce the final model to 9 models and classical examples. 2 Some Delicate Nature of GrC In common practices, very often, we loosely regard any collection of objects as a set. Precisly speaking, this view needs some stipulation; it actually implies implicitly that we are ignoring the interactions among objects. Recall that in set theory, every point is discrete in the sense there are no interactions among elements. However in GrC, one of the manin concerns is the interactions among grnanules, so the collection of granules, called granular structure (GrS), is more than a set of granules. In fact, there are three states for GrS; we will discuss this in the next paper. For now, to understand some of the delicate nature of GrC, let us recall some practices in mathematics. Example 1 Examples from Algebraic geometry: Let Z be the ring of integers. That is, we are considering not only the set of integers but also its two operations, addition and multiplication. A prime ideal is a subset that is closed under addition and mutiplication by any elment of Z Let p be a prime number, then the prime ideal is a subset 2

3 {..., 2p, p, 0,p, 2p,...}, which is closed under addition and multiplication with any element in Z/ In algebraic geometry, each prime ideal is often regarded as a point, then the collection of prime ideals, denoted by Spec(Z), is a set. However, if the interactions among these prime ideals are considered, Spec(Z) is turned into a topological space under Zariski topology. Here are some more elementary examples. Example 2 Let U = {e 0,e 1,e 2,e 3 } be a finite set. 1) from Partition Theory: Let β be the collection {{e 0,e 1 }, {e 2,e 3 }} of subsets. As all subsets are mutually disjoints, β is an honest classical set. Next, we consider a very common example, which often has not been carefully considered. Let β be the collection of all subsets of U. In this case, there are interactions among these subsets as there are overlapping among them. Let us examine how β has been handled. 2) In Set Theory: For casual users, β is often regarded as a set and is called the power set. This is valid only if we disregard the interactions among these subsets. However, 3) In Algebra: β is a Boolean algebra, when we consider the interactions in terms of intersection and union. 4) In Lattice Theory: β is a lattice, if we do consider the union, intersection and incluson together 5) In Algebraic topology: If we do consider the interactions in terms of inclusion only, then β is a partial ordered set, and has a nice goemtrical representation: We take U = {e 0,e 1,e 2,e 3 } to be a set of linearly indepdent points in a Euclidean space. then β can be interpreted as a simplicial complex as follows: 1. U = {e 0,e 1,e 2,e 3 } is a collection of vertices 2. β is a collection of simplexes (a) Four singletons are the four 0-simplexes: i = {e i },i = 0, 1, 2, 3. (b) Six subsets of two elements are the six 1-simplexes: ij = {e i,e j },i < j = 0, 1, 2, 3. (c) Four subsets of three elements are the four 2-simplexes: i = {e i,e j,e k },i < j < k = 0, 1, 2, 3. (d) One subset of four elements is the 3-simplexes: This simplicial complex is called the closed tetrahedron. 3

4 3 Inductive Definition of Granulation The following examples collectively define inductively the concept of granulation. 3.1 Commutative Granules E1 Ancient Practice: Granulation of Daily Objects. Many daily objects are routinely granulated into sub objects. For examples, Human body is granulated into head, neck,...; The surface of earth has been granulated into hills, plateaus, planes... This class of examples are intrinsically fuzzy, vague and imprecise, more precisely, on the boundaries of granules. There are easy solution, but not adequate: One can easily write down a membership function to represent a granule, such as head, neck or body. However, each expert may come up a distinct membership function, and therefore may have a distinct theory of GrC on human body. This is not a satisfactory theory. As there is no unfied view on all of these distinct expert dependent theories. In this paper, a new qualitative fuzzy set theory is used to model this class of examples; they are referred to as 9th GrC model. E2 Ancient Mathematics: Intuitive granulation of the space and time The space and time has been granulated into granules of infinitesimals by early scientists. Of course, mathematically, the notion of infinitesimal granules does not really exist. Nevertheless, it has noisily played a very important role in the history of mathematics. This intuitive notion led to the invention of calculus by Newton and Leibniz. Actually the idea was much more ancient; it was in the mind of Archimedes, Zeno, and etc. Yet the solutions were in modern time. It led to the theory of limit (18th century), topology (early 20 century [30]) and nonstandard analysis, which formally realized the original intuition. (mid 20th century [34])/ The modern theories of this ancient intuition have inspired two models, First GrC Model and Second GrC Model; they have been referred to as neighborhood systems and partial coverings respectively in pre-grc time. E3 Classical Case: Partition (equivalence relation) This class of examples have been well studied in mathematics and and recently in computer science by rough set community. E4 Granules of Uncertainty from Quantum Mechanics Heisenberg uncertainty principle states that, in general, neither the momentum nor the position of a particle can be determined simultaneously with arbitrary great precision. In other words, a great precision of momentum can determine only a probabilistic neighborhood of positions and vice versa. E5 Granules of Knowledge from Computer Security 4

5 In many computer systems,. Discretionary Access Control model assigns each user a set of users(friends) who can access his files. However, we also consider a set of users(foe), who cannot access his files. This is called explicitlky denied list in military security. Examples [E4] and [E5] are real world examples. The idea has been simplified into 3rd GrC Model. It was called binary neighborhood system in pre-grc terminology. Mathematically, it is equivalent to a binary relation. Geometrically, a binary relation is a graph or network. They are the major data structures in computer science. The example [E4] is also in the category of random variables; we are expecting to see it playing heavy roles in future papers, 3.2 Non-Commutative Granules Next, we give examples of non-commutative granules, which are generalizations of partial coverings and binary relations (Second/Third GrC Model). E6 Granules of Knowledge in Data Mining: One of the important concept in data mining (associaiotn rules) is the frquent itemset. It has two views: 1. A frequent itemsets is a collection of constant sub-tuples in a given relation. 2. A relational table (which is called an information table in rough set community) can be viewed as a knowledge representaton of a universe which is a set of entities. In this view, each attribute (column) defines an equivalence relation on the set of entities; this was observed by Z.Pawlak in late 1982 and Tony Lee in early 1983 [7], [32]. An attribute value can be regarded as the name of an equivalence class. In this view, a frequent itemset is an intersection of equivalence classes, whose cardinal number is large than a given threshold. This intersection is a granule [?]. E7 Granules of Knowledge in Web/Text Mining. High frequent sets of co-occurring keywords (fsck) in document set or the web can be regarded as an abstract ordered simplex. Moreover the apriori principle of frequent item sets turns out to be the closed condition of simplicial complex [?]. Recall that a simplicial complex (of keywords) consists of two objects: one is a finite set of vertices (keywords). Another one is a family of ordered/oriented/unordered subsets, called ordered/oriented/unordered simplexes that satisfy the closed condition, namely, any subset of a simplex is a simplex. This is an important mathematical structure in algebraic/combinatorial topology. Currently, it is finding its way to web technology [23], [?], [25]. E8 Granules of Knowledge from Social Networks. 5

6 The collections of committees in a human society (a set of human beings) is a granulation. Observe that each member may play distinct role in a committee, so the members cannot exchange their roles freely. We can view the collection of roles as a relational schema. If we do so, then a committee is a tuple. Observe that different types of committees have different schema. The set of committees under the same schema forms a relation. A collection of n-nary relations for various n is a granulation of the society. The example [E6], [E7], and [E8] are modeled inti Fifth GrC Model. 3.3 Granules of Advanced Objects Roughly, the examples [E1] to [E8] given above are granulation of data Now we will turn to more complicated objects. First, we observe that all examples can be fuzzified (type I fuzzy set), which is fully characterized by a membership function. Mathematically, they are bounded functions. Moroever, the example [E4] in Heisenbeerg Uncertainty Principle is actually a granulation of random varibles (measurable functions). Hence, it should be nature to extend the consideration of membership functions and measurable functions to general functions. So we include the following example into our collections E9 Functional Granulation: Radial-Basis-Functions is a granulation in some functin space [31] More generally, a collection of functions (e.g. Radial-Basis-Functions)that satisfies the universal approximation peoperty will be regarded as functional granulation. In fact, we will extend it to measures/probabilites and generalized fucntions This class of examples led to Sixth GrC model. E10 Computers or clusters of computers in Grid/Cloud computing are granules. These are hardware examples. Traditionally, How to solve it [33] has not been any part of formal mathematics, however, how to compute is an integral part of computing. So GrC includes some mathematical/computational practices. E11 A granule can be a subprogram in a program, or a lemma in a mathematical proof, when the proof is computable. Formally, within the computable domains, such a granule is a sub-turing machine. The so called Divide and Conquer in computer science is actually in a granulation of Turing machines This idea is modeled in 7th GrC Model. A lemma in a mathematical proof is a granule conceptually, however, we should not include it in GrC unless the mathematical proof itself is computable. However, in this class, we may extend the idea of 7th GrC Model to GrM.. Let us sumarize this section by Zadeh s Intuitive Definintion. 6

7 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. [41], [17]. We will praaphrase into 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 some constraints or forces, such as indistinguishability, similarity or functionality.. 4 Formal Definition of Granulation In GrC2008, I proposed to use the category theory based model (Eighth GrC Model) as the final formal model for GrC. To state it, it requires some category theory. As the category is not a common sense in computr science, we will introduced some simpler models as stepping stones. 4.1 Some Simpler Models First, we shall take Zadeh s clump of objects as a set, and we propose: 1) 2nd GrC Model. Let U be a classical set, called the universe. Let β = {F k k K} be a family of subsets. Then the pair (U,β), is called the 2nd GrC Model. If we use pre-grc terminology, β should be called a Partial Covering(PCov). However, most of those papers adressed only on (full) Covering. Partial covering is a special case of NS, in this sense, they were covered in Pre-GrC time. In this model, we have implicitly assumed that the constraints or forces are uniformly exerted on each member of the granule. So a granule is a set. If the constraints or forces are not uniform, one can regard them as a schema (of a relational database). In this case the collection of granules are tuples of some relations. These are modeled in 5th GrC model, Let us introduced a convention: Convention for index. In computer science, the index often runs through a countable set. In this case, we often denoted it as follows: k = 1, 2,...k,... without naming an index set. As this paper will include GrM, we will take the following convention: the lower case letter, say k, denotes the parameter that runs though a set, K, whose name is the corresponding cap letter. To make the understanding of the 5th GrC model easier, we explain the generalization process as follows: 1. U is generalized to U = {U h j h H;j J} 7

8 F k U is generalized to R k {U k l k K b H;L k } ) 5th GrC Model 1. Let U = {U h j h,h H,j h J h } be a given family of classical sets, called the universe. Note that distinct indices do not imply the sets are distinct. 2. Let {U k l k K b H;L k b J k } be a family of Cartesian products of various lengths n, where n = L k is the cardinal number of L k that may be infinite. Here b means a sub-bag. Recall that a bag is a set that allows the repetition of some elements [4]. 3. Recall that an n-ary relation is a subset R m of a product space in the previous item. 4. Let β = {R m m M} be a given family of n-ary relations given in previous item for various n; note that n can be infinite. Then the pair (U,β), called Relational GrC Model, is the formal definition of Fifth GrC Model 4.2 Category Theory Based Models First, we would like to observe that 8th GrC Model is abstractly the same as the category of relational databases [13]. In other words, from the point of views of mathmatical structures the category of data and knowledge are the same. However, their meaning are very different. Let us set up some language for Category Theory. Definition 1 A category consists of 1. A class of objects, and 2. For every ordered pair of objects X and Y, a set Mor(X,Y ) of morphisms with domain X and range Y ; if f Mor(X,Y ), we write f : X Y 3. For every ordered triple of objects, X, Y and Z, a function associating to a pair of morphism f : X Y and g : Y Z their composite g f : X Z These satisfy the follwoiing tow axioms (a) Associativity. If f : X Y, g : Y Z, and h : Z W, then h (g f) = (h g) h : X Z 8

9 (b) Identity. For every object Y there is a morphism I Y : Y Y such that if f : X Y, then I Y f = f and if h : Y Z, then h I Y = h If the class of objects is a set, the category is said to be small. Here are some examples. 1. The category of sets: The objects are classical sets. The morphisms are the maps. 2. The category of fuzzy sets: The objects are fuzzy sets (of type I). The morphisms are the maps. 3. The category of crisp/fuzzy sets with crisp/fuzzy binary relations as morhisms: The objects are classical sets. The morphisms are crisp/fuzzy binary relations. In crisp case, this is the Category of Entity Relationships Models. 4. The category of power sets: The object U X is the power set P(X) of a classical set X. Let U Y be another object, where Y is another classical set. The morphisms are the maps, P(f) : U X U Y that are induced by maps f : X Y. 5. The category of topological spaces: The objects are classical topological spaces. The morphisms are the continuous maps. 6. The category of neighborhood system spaces (NS-space): The objects are NS-spaces; see First GrC model.. The morphisms are the continuous maps. Let CAT be a given category; we adopt the index convention stated above. Definition 2 Category Theory Based GrC Model: 1. C = {C h j h H,j h J h } be a family of objects in the Category CAT. 2. There are families (which are bags) of Cartesian products {C k l k K b H;L k b J k } of various lengths n, where n = L k is the cardinal number of L k that may be infinite. They are called product objects. 3. An n-ary relation object R j is a sub-object of a product object. 4. β be a family of n-ary relations (n is any cardinal number and could vary). The pair (C, β), called Categorical GrC Model (Eighth GrC model), is the formal model of granulation. By specifying the general category to various special cases, we have all the models. We will explain the specializations in the follwoing few sections We give a summary here: Proposition 1 The 8th GrC can be specified to 9 models as follows: 1. Models of Non-commutative Granules (a) By taking CAT to be the category of qualitative fuzzy sets, we have 9th GrC model (b) By taking CAT to be the category of Turing Machines, we have 7th GrC model 9

10 (c) By taking CAT to be the category of fuzzy sets(membership functions), functions, random variables, and generalized functions, we have 6th GrC model (d) By specifying the category to be the category of sets, we have 5th GrC model. 2. Models of Commutative Granules (a) By limiting the product objects to be product of two objects in 5th GrC model, we have 4th GrC Model. (b) By limiting the number of binary relations in 4th Grc Model to be one, we have 3rd GrC Model. (c) By requiring all n-nary relations to be symmetric, we have 2nd GrC Model (d) The reduction to 1st GrC Model is treated in Section Overview of GrC Models Schematically we summarize the relationships, based on the Granular Structures, of early GrC Models as follows: (the diagram will be different, if it is based on their approximation spaces) is a two way generalization but they are not inverse to each other.,, and are one way generalizations. GM means GrC Models and RST means Rough Set Model. 5th GM 8th GM 4th GM 3rd GM 2nd GM 1st GM 3rd GM RST 8th GM 7th GM 9th GM 5 Neighborhood System - 1st GrC Model The ancient intuitive notion of infinitesimal granule, [E2] in Section 3, has been formalized in two ways: 1. The formal infinitesimal granule in non standard world (NSW). 10

11 2. Topological Neighborhood System (TNS) in standard world. We will focus on TNS. It is important to observe that the ancient intuition of infinitesimal granules (with the required properties) is formalized, not by a set, but by a family TNS(p) of subsets, that satisfies the (local version) axioms of topology. Nevertheless, in this paper a (modern) granule will refer to a neighborhood, but, not to the whole neighborhood system. Definition 3 The notion of topology can be defined in two equivalent ways: 1. Global Version: A topology τ is a family of subsets, called open sets, that satisfies the axioms of topology: τ is closed under finite intersections and arbitraty unions of τ. 2. Local Version: A topology, called topological neighborhood system (TNS), is an assignment that associates each point p a family of subsets, TNS(p), that satisfies the following axioms of topology: (a) If N TNS(p), then p N, (b) If N 1 and N 2 are member of TNS(p), then N 1 N 2 TNS(p) (c) If N 1 TNS(p) and N 1 N 2, then N 2 TNS(p), (d) If N 1 TNS(p) then there is a member N 2 TNS(p) such that N 2 N 1, and N 2 TNS(q) for each q N 2 (that is, N 2 is a neighborhood of each of its point), These two definitions lead us to First and Second GrC Models (Local and Global GrC Models). Let U and V be two classical sets. Let NS be a mapping, called neighborhood system(ns), NS : V 2 (P(U)), where P(X) is the family of all crisp/fuzzy subsets of X. 2 Y is the family of all crisp subsets of Y, where Y = P(U). In other words, NS associates each point p in V, a family NS(p) of crisp/fuzzy subsets of U. Such a subset is called a neighborhood (granule) at p, and NS(p) is called a neighborhood system at p. Definition 4 The 3-tuple (V,U,β) is called First GrC Model (Local GrC Model), where β is a neighborhood system (NS). If V = U, the 3-tuple is reduced to a pair (U,β). In addition, if we require NS to satisfy the topological axioms, then it becomes a TNS. Proposition 2 Let (V,U,β) be a 4th GrC Model, where β = {B i i I} is a collection of binary relations B i on V U. Let B i (p) = {(p,x) x U} and p V. Then NS(p) = {B i (p) ı I} defines a neighborhood sysgtem on p. Let p vary through V, we have an NS and (V,U,NS) is a 1st GrC Model. 11

12 5.1 The Concept of Near and Contexts The following arguments are adopted from my pre-grc paper [16] The notion of near is rather difficult to formalize. Let us examine the following examples. 1. Is Santa Monica near Los Angels? Answers could vary. For local residents, who have cars, answers are often yes For visitors, who have no cars, answers may be no 2. Is 1.73 near 3? Again answers vary; they depend on what was the agreement on the tolerance radius, in other words, in the given context. Intrinsically near is a subjective judgment. One might wonder whether there is a scientific theory for such subjective judgments? Mathematical analysis has offered a nice solution. They simply include all possible contexts into its formalism. Here is the formalism of the second question: Given the radius of an acceptable error, say, radius of errors 1/100 (a selected context) Is 1.73 near 3? With this context selection, that is, 1/100 is acceptable error, then 1.73 is near 3! On the other hands, if the context (agreement) has changed to 1/1000, then 1.73 is NOT near 3! Clearly, in this numerical example, the collection of all possible contexts is the collection of all positive real numbers. We often use ɛ to denote the variable that varies through such a context set. Similarly, let us assume a neighborhood system has been assigned to each city in Los Angeles area: For example, based on car driving, public transportation, walking and etc, we assign a neighborhood to each city for each context. Under such a concept of neighborhood system, the question 1 above can be formulated properly as follows: Assuming that we have selected context(taking public transportations) Is Santa Monica near Los Angels? Now we can have s definite answer to this question. So a neighborhood system is a good infrastructure for addressing the concept of near! These analysis leads to the following conclusions. 1. In Modeling, a neighborhood system is a good infrastructure for providing all possible contexts. 2. Under this model, in an application, selecting a context means selecting a fixed neighborhood as a unit of tolerance(uncertainty). Now, based on such a concept, we re-examine previous examples Example 3 If we have chosen driving half an hour as acceptable distance of Near, then Santa Monica is near Los Angels. 12

13 Example 4 Let the collection of ɛ-neighborhoods at each point be the neighborhood system of the real numbers R; Then(R, ɛ-neighborhoods) provides the proper contexts for discussing near answers( approximate answers), where ɛ could take any real value. (R, ɛ-neighborhoods) is a First GrC Model [?] Now, we re-state the previous example using this First GrC Model 1. Assuming we have agreed ɛ = 1/100 is acceptable, then 1.73 is near 3 2. But, if we have only agreed ɛ = 1/1000 then 1.73 is not near 3 3. Next let us consider a deeper question Is the sequence 1, 1/2, 1/3,.., 1/n,... near zero? By near zero we mean: For any given ɛ > 0 (a context at zero), there is a number N = [1/ɛ] + 1, such that, ɛ > 1/n for all n > N., where [ ] is the integer part of. Such a concept of near for all contexts is said to be absolutely near. For readers who familiar with the standard (ɛ, δ)-definition of limit can spot the origin of neighborhood systems. Such a context free (all possible contexts) answer is precisely the classical notion of limits, lim n 1/n = 0. Using our language, we may say that limit is the context free answers of near Perhaps we should also point out here that there is no context free answers for the question whether two points are near. A Lesson 1. Each granule provides a context, a state or an agreement as what to be considered as near, 2. Granulation, namely beta, provides the complete contexts/states that can be used in reasoning about near-ness. Brief pre-grc historical notes: 1. In , Lin generalized TNS to the Neighborhood Systems(NS) by simply dropping the (local version) axioms of topology [9], [11] and apply it to approximate retrievals. Each neighborhood was treated as a unit of uncertainty. 2. In the same year (1989), Lin also examined a non-reflexive and symmetric binary relation (conflict of interests) for computer security from the view of NS [10]. 3. Abstractly, Lin imposed NS structure on the attribute domains for approximate retrieval. Taking this view, we should mentioned that earlier D. Hsiao imposed equivalence relations on the domain for access precision in early 1970 [3], [38]. In 1980, S. Ginsburg and R. Hull had imposed partial ordering on attribute domains [5], [6]. 13

14 4. In much earlier, NS was studied in [35] as a generalization of topology. Note that however, there are fundamental differences, for example, the concept of closures are different. The term pre-topology also has been used for referring NS and TNS. 5. In early GrC period, Lin, by mapping the NS onto Zadeh s intuitive definition, used NS as his first mathematical GrC model [17], [18], [19]. 6 Second GrC Models and Modern Examples As in the previous case, by dropping the global axioms of topology, we have Second GrC model. Definition 5 Second GrC Model: The Pair (U,β), where β is a family of subsets of U, is called Global GrC Model. The β, some time, is referred to as a partial covering(pcov). Note that Second GrC model is a special case of First GrC model: If we regard the sub-collection of all members of the partial covering β, that contains p, as a neighborhood system at p, then this Second GrC model is an example of First GrC model. The modern example, simplicial complexes, is an important example of such a model: A simplicial complex consists of a set of vertices and a family of subsets, called simplexes, that satisfies the closed condition [36] [Digression] Perhaps, it is worthwhile to note that the closed condition of simplicial complex is the apriori principle in association (rules) mining. This observation play an important role in document clustering [25]. 7 Third and Fourth GrC Models and Modern Examples In this section, we will build a new model that realized modern example [E4] and [E5]. Recall that [E4] concludes that, a precise measure of the momentum can only determine a (probabilistic) neighborhood of positions; and [E5] concludes that in computer security, the Discretionary Access Control Model (DAC) assigns to each user p a family of users, Y i, i = 1,..., who can access p s data. In other words, each p is assigned a granule of friends. To formalize these examples, let U and V be two classical sets. Each p V is assigned a subset, B(p), of basic knowledge (a set of friends or a neighborhood of positions ). p B(p) = {Y i, i = 1,...} U Such a set B(p) is called a (right) binary neighborhood and the collection {B(p) p V } is called the binary neighborhood system (BNS). Definition 6 Third GrC Model: The 3-tuple (U,V,β), where β is a BNS, is called a Binary GrC Model. If U = V, then the 3-tuple is reduced a pair (U,β). 14

15 Observe that BNS is equivalent to a binary relation(br): BR = {(p,y ) Y B(p) and p V }. Conversely, a binary relation defines a (right) BNS as follows: p B(p) = {Y (p,y ) BR} So both modern examples give rise to BNS, which was called a binary granular structure in [17]. We would like to note that based on this (right) BNS, the (left) BNS can also be defined: D(p) = {Y p B(Y )} for all p V }. Note that BNS is a special case of NS, namely, it is the case when the collection NS(p) is a singleton B(p). So the Third GrC Model is a special case of First GrC Model. The algebraic notion, binary relations, in computer science, is often represented geometrically as graphs, networks, forest and etc. So Third GrC Model has captured most of the mathematical structure in computer science. Next, instead of a single binary relation, we consider the case: β is a set of binary relations. It was called a [binary] knowledge base [17]. Such a collection naturally defines a NS. Definition 7 Fourth GrC Model: the Pair (U,β), where β is a set of binary relations, is called Multi-Binary GrC Model. This model is most useful in data bases; hence it has been called Binary Granular Data Model(BGDM), in the case of equivalence relations, it is called Granular Data Model(GDM) Observe that a Fourth GrC Model can be converted, say by a mapping G, to a First Model. Conversely, a First GrC Model induces, say by F, to a Fourth Model. So First and Fourth models are equivalent, but not naturally, namely, G and F are not inverse to each other. 8 Models for Further Examples As we have observed in Section?? that the collection of n objects that are drawn together is, not necessary a subset, but is a tuple in an n-ary relation. For example, if the universe is a human society then a group of people may be drawn into a committee with distinct roles, such as the chair, vice chair, secretary, treasurer, and etc. As every member has different role, they can not be swapped around. So the committee is not a set; it is a tuple under the schema that consists of distinct roles. Definition 8 5th GrC Model: 1. Let U = {U h j,h,j, = 1, 2,...} be a given family of classical sets, called the universe. Note that distinct indices do not imply the sets are distinct. 15

16 2. Let U j 1 U j 2... be a family of Cartesian products of various length. 3. Recall that an n-ary relation is a subset R j U j 1 U j 2...U j n. 4. Let β = {R 1,R 2,...} be a given family of n-ary relations for various n. The pair (U,β), called Relational GrC Model, is a formal definition of Fifth GrC Model Note that this granular structure is the relational structure (without functions) in the First Order Logic, if n only varies through finite cardinal number For next two models, we will use the language of category theory in next sub-section. We may note that we have not committed ourselves to every specific details yet. Definition 9 Sixth GrC Model is in the categories of functions, random variables and even generalized functions. Fuzzy sets are described by membership functions, so granules can be regarded as membership functions; note the First to 5th GrC Models include fuzzy sets. Hence, we consider further generalizations: granules are functions, random variables (measurable functions) generalized functions (e.g. Dirac delta functions). In the case, a granule is a function, we may require that the granular structure (the collection of granules) has the universal approximation property, namely, any function in the universe can be approximated by the functions in the collections. The membership functions selected in fuzzy controls do have such properties. In neural networks, the functions generated by the activation functions also have such property [31] In the case of probability/measure theory, quantum mechanics may be a good guiding example. Definition 10 Seventh GrC Model is in the category of Turing machines. For examples, a collection of lemmas in mathematical proof (mechanizable), a set of subprograms in a computer program, or a computer or cluster of computers in grid/cloud computing are granules in the model. Definition 11 Ninth GrC Model is in the category of qualitative fuzzy sets. This model was proposed after the Eighth GrC model. It has not been published in printing form yet. The idea is similar to the model that we have called it sofset(this is not a typo) [15]. It associates to each real world fuzzy set, a collection of membership functions; please watch for new development. 16

17 9 Qualitative Fuzzy Sets - A Proposal Fuzzy theories have been driven by applications. Naturally implicitly or explicitly, some contexts of their specific applications may have imposed into their theoretical frameworks. In this paper, we will try to deveop a fuzzy set theory that are independent from any particular context Intuitively, a real world fuzzy set should be represented by an elastic membership function. What is the intuition behind the term elasticity? We believe it is a set of membership functions, each of which represents a state of the real world fuzzy set (in different time or situation). So A Proposal: 1. A real world fuzzy set should be represented by a granule of memberhsip functions. 2. Each membership function (in the granule) represent a particular state/contexts of the real world fuzzy set 3. The granule.provides the complete states/contexts that can be used in reasoning about real world fuzzy set. Kandel is the first one to use more than one membership functions to represent a single real world fuzzy set of very large numbers. We view it as two states/contexts of the real world fuzzy set. We suggest the readers to detour to Section 11 for a quick glance on the various notions of sofsets (not a typo). Definition 12 Ninth GrC Model is defined as follows: 1. MF(U) is the membership function space on the universe U. 2. Each membership function space in MF(U) represents a specific view of some real world fuzzy set. In other words, selecting a membership function(similar to selecting an ɛ) is equivalent to choosing a state/context 3. Ninth GrC model (MF(U),β) is a First GrC Model on membership function space MF(U) such that (a) β is a NS on MF(U) and is a Boolean Algebra under the union and the operation circ defined below (Section 9.1). (b) Each neighborhood N β, is a collection of membership functions that represents a real world fuzzy set, called a qualitative fuzzy set. (c) Each member of N represents a state/context of a rel world fuzzy set Example 5 Let C be a covering (as defined in Section 11), let β be the Boolean algebra generated by unions and intersections of C. Then a member C of β (a granule) represents a real world fuzzy set. The granule C is a called a qualitative fuzzy set; each membership function in C represent a state of the real world fuzzy set. 17

18 9.1 Operations in Neighborhood Systems Let us consider the NS in a First GrC Model. It is clear a Ninth GrC Model is a special First GrC Model. The key question is: Can we generate a Ninth GrC Model from the NS? Here is the required constructions: Let NS(p) be the neighborhood system at p of a First GrC Model; Let N(p) represent an arbitrary neighborhood of NS(p) Let C N (p), called the center set of N(p), consists of all those points that have N(p) as its neighborhood. (Note C N (p) is called center set; in the case of topological spaces, it is the maximal open set in a topological neighborhood). Now we will observe something deeper: Let G(p) be the collection of all finite intersections of all neighborhoods in NS(p). Then a hard question is: Does the intersection of two neighborhoods (in G(p) and G(q) at distinct points p and q, belong to some G(r)? Proposition 3 The theorem of intersection 1. N(p) N(q) is in G(p)=G(q), iff C N (p) C N (q). 2. N(p) N(q) is not in any G(p) p, iff C N (p) C N (q) =. In GrC, we regard N(p) as a known basic knowledge, and we defined the knowledge operations [?]: Let be the and of basic knowledge. For technical reasons, the is regard as a piece of the given basic knowledge. Definition 13 New operations 1. N(p) N(q) = N(p) N(q), iff C N (p) C N (q). 2. N(p) N(q) =, iff C N (p) C N (q) =. Observe that BNS is a special cases of NS So we have. Definition 14 Let B be a BNS, then 1. B(p) B(q) = B(p) = B(q), iff C B (p) = C B (q). 2. B(p) B(q) =, iff C B (p) C B (q) =. Note that B(p) B(q) may not empty, but it is not a neighborhood of any point. Observe that in Binary GrC Model, two basic knowledge are either the same or the set theoretical intersection does not represent any basic known knowledge. Theorem 1 Given a First GrC Model (U,NS), then the Boolean algebra β generated from NS using operation forms a Ninth GrC Model (U,β) Theorem 2 If NS is a covering C, then is the intersection. So the Boolean algebra β is the Boolean algebra generated by C using unions and intersections. 18

19 10 Conclusions In Springer Encyclopedia, we have proposed a category theory based model, as the Formal Model for GrC. We said the model can be specialized into various category to realize all the classical examples, including the first example, the granulation of human body. However, the claim on the realization of the first example was not in printing form This paper is to realize this claim, and we have shown that 1. The universe of discourse is the domain of interests. In this paper, the domain of interests of collection of qualitative fuzzy sets. Taking this view, the Ninth GrC Model (MF(U),β) is the universe of discourse. 2. The union and intersection of qualitative fuzzy sets are provided by the Boolean Algebraic Structure on the collection of granules. Note that the intersections are actually the knowledge operation. 3. A granule consists of a collection of membership functions is regarded as a representation of an elastic membership function of a real world fuzzy set. Elasticicity is dynamic, so to represent a dyanmic system, we have to use a trajectoiry of states to represent. So the granule is the trajectory conists of all the current states. 4. Each membership function in the granule represents a state of this elastic membership function 11 Appendix-Sofsets and Fuzzy Sets Let us recall some results from [15] with some serious revision. Membership function space will be the focal points. To avoid monotonous, we will use terms, such as, space, family, and collection as synonym of crispy set. A space often refers to a very large set; a collection or a family often refers to a set of sets. Definition 15 Let U be a given classical set, called the universe, and let FX : U M be a map, where M is, in general, a membership space [?]. FX is called a membership function; FX(x) is called the grade or degree of membership of x U. If M is the set of two elements 0, 1, then FX is the characteristic function of a classical crispy set. If M is a unit interval, then FX is the membership function of a classical fuzzy set. We will focus on classical fuzzy sets. From now on M will be the unit interval [0, 1]. The collection of all FX s is called the membership function space on U, denoted by MF(U). As in [15], let C be a neighborhood system on a membership functions space. Each neighborhood represents a qualitative fuzzy set; we shall call it sofset (- not a typo) Definition 16 A member of C is a: 19

20 1. W-sofset (Weighted Soft Set), if C consists of singletons. A membership function is treated as a characteristic function of a soft set. This is essentially Type I fuzzy set. 2. F-Sofset (Finite-multi Soft Set), if C consists of finite sets only. 3. P-Sofset (Partitioned Soft Set), if C forms a crispy partition. This is mathematically a most beautiful theory. Realizing that a fuzzy set (that tolerates perturbation) has to be represented by a set of membership functions. Each set represents one and only one fuzzy set. Then the space of membership functions is partitioned into equivalence classes. So P-sofset theory is very elegant and beautiful. However, one may wander how could there be a natural partition in a continuous membership function space. This lead us to a more general point of view, namely, the next few items. 4. B-Sofset (Binary Neighborhood Soft Set), if C forms a binary neighborhood system (i.e., the neighborhood system is defined by a binary relation). Binary neighborhoods are geometric view of binary relation. Intuitively, related membership functions are geometrically near to each other. A binary neighborhood system is equivalence to an abstract binary relation, 5. C-Sofset(Covering Soft Set), if C forms a covering. 6. G-Sofset(G-covering Soft Set), if the covering C forms a semi-group under intersection (G may contain empty set) 7. N-Sofset (Neighborhood Soft Set), if C forms a neighborhood system (NS); see Section??.. This is the target concept, qualitative fuzzy set. It is most general case; it contains all previous cases. 12 Future Directions Granular Computing is still in its inception stage; possible directions are wide open. Here we will focus only on those issues that are touched in this article. 1) Developments of Categories In this paper, a category based model is proposed as the Formal Model for GrC. It can be specialized into various models to realize all the classical examples, including the first example, the granulation of human body. We should note that the claim on the realization of the first example is not in printing form yet. However, this author feels that it is important to inform the readers that is occurring. The key to realize the first example is based on the category of qualitative fuzzy sets or sofsets(this is not a typo); please watch for new development. The categories of functions, random variables(measurable functions) and Turing machines are also need to be developed. 2) Developments of Granular Structures 20

21 Given a granular structure, we associate it with four structures (including itself). Among them quotient and knowledge structures are mathematical consequences of granular structure (if it is given mathematically). However the linguistic structure is not a mathematical formalism, but is a natural language formulation. In this paper, there is no report on this direction. We urge the readers to read Zadeh s article. 3) Imported Concepts For information integration (this may correspond to Zadeh s term, organization ), we have illustrated the idea imported from homological algebra. It is unclear if we have imported the correct thinking; but it does point out essential problems in granulate and conquer. 4) GrC and RST RST has been served as the model of GrC developments. So there are a lots of similarity, here, we would like to caution the readers that, there are fundamental differences. For example, the fundamental views of uncertainty are quite different; Pawlak used unable to specify as the base of uncertainty, while GrC regard a granule as a unit of uncertainty (such as uncertainty in quantum mechanics) Also the approximation theories are different. Of course, there are other differences; we skip. 5) GrC, Databases and Data Mining As we have pointed out that the categorical structures of databases and GrC are similar; at the same time, we need to point out the differences in semantics. Nevertheless, we are looking forward to the transfer of database technology to GrC. For data mining, please see database section on the articles by this author on deductive data mining using GrC, and mining decision rules using RST. 6) GrC and Fuzzy Logic Most of expositions have been based on classical sets (and fuzzifiable concepts) For more intrinsic fuzzy view, we strongly recommend the readers to read Zadeh s article. 7) GrC and Clouding Computing Theoretically, cloud computing can be related to the GrC on the category of Turing machines. We expect some strong interactions in near future. As we have observed that GrC is deeply rooted in human thinking, we expect GrC will have many interactions with wide variety of areas. 21

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