Granular Computing II:

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1 Granular Computing II: Infrastructures for AI-Engineering Tsau Young (T. Y.) Lin, Member, IEEE, Abstract What is granular computing? There are no well accepted formal definitions yet. Informally, any computing theory/technology that involves elements and granules (subsets or generalized subsets) may be called granular computing (GrC). Intuitively, elements are the data, and granules are the basic knowledge. So granular computing is the infrastructures for AI- Engineering: uncertainty management, data mining, knowledge engineering, and learning. Index Terms granular computing, data, knowledge, learning, neighborhood system, topology. I. INTRODUCTION What is granular computing? There are no well accepted formal definitions yet. Informally, any computing theory/technology that involves elements and granules (generalized subsets) may be called granular computing (GrC). Intuitively, elements are the data, and granules are the basic knowledge. So granular computing is the infrastructures for AI-Engineering; it includes uncertainty management, data mining, knowledge engineering, and learning.. A. How did it start? What was the initial idea? In the Fall 1996, Professor Lotfi Zadeh recommended granular mathematics to be my area of research during the sabbatical leave at Berkeley. After some consideration, I proposed to restrict the scope to a subset and named it granular computing (GrC) [25]. My first reaction to the term granular mathematics is that it is some sort of uncertainty mathematics. Earlier, we have studied approximate retrieval [6], [1] in which we had attached every point a set of neighborhoods, each of which serves as a generalization of radius of errors. [9] So the neighborhood system came in as a first natural model for granular computing from the prospect of uncertainty and quickly find its way into data mining and learning Interestingly, once the term was changed to granular computing, the notion of divide and conquer slipped into the picture, and is extended to granulate and conquer. So a component of problem solving was added into granular computing. We granulate human body into neck, head, body and etc. This addition was very natural; dynamic programming is not-clean divide (granulate) and conquer. Tsau Young Lin is with Department of Computer Science, San Jose State University. Another result, prio to GrC, but related to, is a mathematical model, called rough logic government, in which we integrated the hard (differential geometric method on non-linear control) and soft computing into a formal model of fuzzy control [10]. This model is a consequence of fuzzy granulate and conquer. II. MATHEMATICAL MODELS FOR GRANULAR COMPUTING Either from the granules of uncertainty or granulated subproblems, we need to understand the granulation or the granulated space. A. Granulation using Classical Sets A good source of inspiration is the notion of infinitesimals: every standard number is determined up to a granule of infinitesimals. There are two classical formalizations for such a notion. One is Abraham Robinson s non-standard analysis, the other is topological space. The notion of limit (recall Cauchy s Given ɛ, there is δ... ) is based on the concept of topology. We will try to use the idea of these formulations to capturing the essence of granulation. First, we will adopt the idea from topological spaces. A topological space is often defined as a pair of objects that consists of one classical set and one family of open sets that satisfies certain axioms. We will take this scheme and propose to define Definition 1: A set based granular model is defined to be a pair (U, β), where U is the universe of discourse, called global granular space and β is a family of subsets X j,j J, called basic granules or elementary granules just to be in sync with rough set theory [22], where J is an index set. Intuitively, U is the data and granules are the basic knowledge about the data and are the building block of (background) knowledge. In the case of uncertainty, granules are the core subsets that have no adequate information. If we put the interpretations away, and back to mathematics, then granules, either from uncertainty or background knowledge, are mathematically the same; we will so proceed. Example 1: 1) If β satisfies the properties of open sets, global granular space is the topological space [23], [21]. 2) If β is a partition, the granular space is a clopen topological space and is called an approximate space in rough set theory. Pawlak believes classification represent human knowledge, so, if β consists of several partitions, the granular space is called knowledge base [22]. As knowledge base often has different meaning, in data /06/$ IEEE 2

2 mining area, we have been called it granular data model [16], [17]. We will illustrate the model with a more delicate example: Example 2: Let us assume U consists of 4 points; see the first tetrahedron in Figure 1 below. β is organized as follows: 1) X 1 4 consists of four elements {a, b, c, d}; It can be thought of as a set of linearly independent 4 points in Euclidean space. They determined an open tetrahedron (a open simplex of dimension 3). 2) Each one of X 1 3, X 2 3, X 3 3, and X 4 3 consists of three element. Each can be thought of as a set of open triangles (2 dimensional faces) of the tetrahedron. 3) Each one of X 1 2, X 2 2, X 3 2, X 4 2 X 5 2, and X 6 2 consist of two elements. Each is a simplex of dimension 1. 4) Each one of X 1 1, X 2 1, X 3 1, and X 4 1 consists of one elements. Each is a simplex of dimension 0. 5) This collection of all these simplexes above form a simplicial complex. B. Granulation using Relational Structures The Example?? leads to further consideration. In algebraic topology, we do consider ordered simplexes. So each X j may be ordered. In other words, X j is a tuple of some relation. Let us group the tuples of the same size together. Each group forms a relation. So a granular space is a set with many relations; this is essentially the Tarski s relational structure (without functions) in first order logic; we have reach the same model as in [19]. Definition 2: A global granular model is defined to be a pair(u, β), where U is the universe of discourse, called global granular space and β is a family of n-ary relations X j,j J(n =1, 2,...). Each tuple is called a basic granule. Each relation represents some hidden semantic force that organize a set of granules (basic knowledge). Proposition 1: If all relations are symmetric (order independent), then Definition 2 and Definition?? is the same. C. Granulation using Neighborhoods Topological space can be defined globally or locally. We have given a global definition, now we will give a localized version with slight generalization (to two universes) Definition 3: A neighborhood granular model: Let V and U be two universes (classical sets); they could be the same universe. For each object p V, we associate a family of subsets, denoted by NS(p) P(U), that is, we have amapb:v 2 P (U) : p NS(p), where P(U) means the power set of U, and 2 P (U) means the power set of all subsets. Note that P(U) could include all classical or fuzzy subsets; but 2 U is restricted to the set of all classical subsets. NS(p) is called a neighborhood system at p. NS(U)={NS(p) p V } is a neighborhood system of V. D. Granulation using Binary Relation In the case NS(p) is a singleton, we define Definition 4: A neighborhood granular model: The map B: U B(p) is called a binary granulation (BG) and the collection {B(p) p U} a binary neighborhood system (BNS). Note that B(p) could be an empty set. Note that q B(p) do not imply B(p) is a neighborhood of q, in the case V=U. Definition 5: It is clear that given a map B gives rise to a binary relation BR V U and vice versa: BR = {(p, x) x B(p) and p V }. Proposition 2: 1) 2) BR, BNS, and BG are equivalent and will be treated as synonyms. E. Granulation using Generalized Sets The most well known generalized sets are fuzzy sets. Mathematically a fuzzy set is a unit-interval valued function, but is interpreted as a generalized set. A function value f(x) at x are interpreted as the degree of membership of x. Zimmerman has generalized it to bounded real valued function [26], we will extend it to any real value function. In this paper A fuzzy set and a real value function are synonym. So we may extended the scope of granules from classical sets to generalized sets. Namely the basic granules now can be classical sets or generalized sets (functions). To widen the class of readers, we may use the term functions, instead of fuzzy sets; however the intuitive connection to classical sets is fuzzy set based Definition 6: Let U be a classical set. But β is a family of functions (fuzzy sets). The pair (U, β) will be a functional/fuzzy granular model. Further, we say the model is a universal approximator in Function Space F, if all finite linear combinations of β is dense in F (in some topology). III. UNCERTAINTY AND GRANULAR COMPUTING The basic notion here is: to each point we attach an uncertainty region, such as radius of errors. In other words, within the radius of error, all objects are considered equivalence. Essentially, we are applying neighborhood granular model to approximate retrieval: To each point we have attached a neighborhood of uncertainty, namely, all objects within the neighborhood are regarded as mutually replaceable. So we replace a given query automatically to nearly equivalent one. The idea is essentially in [6], [8];it was summarized in [12]. A. Uncertainty and Approximate Queries Let us consider the following Restaurant Database. For each attribute there are four neighborhood neighborhoods: veryclose-neighborhood(vcn), close-neighborhood (CN), farneighborhood(fn), and very-far-neighborhood(vfn). These four binary neighborhoods on each attribute are given in: 1) The binary neighborhood system of LOCATIONattribute: 3

3 RESTAURANT TYPE LOCATION PRICE ACTION Wendy American West wood inexpensive keep Le Chef French West LA moderate raise GreatWall Chinese St Monica moderate may-raise Kiku Japanese Hollywood moderate raise South Sea Chinese Los Angeles expensive may-raise TABLE I ARESTAURANT DATABASE a) VCN W estwood = {Westwood, Santa Monica, West LA} b) CN W estwood = {Westwood, Santa Monica, West LA, Hollywood, Los Angeles} c) FN W estwood = {Westwood, Santa Monica, West LA, Hollywood, Los Angeles, San Francisco} d) VFN W estwood = { Westwood, Santa Monica, West LA, Hollywood, Los Angeles, San Francisco, New York City} e)... f)... 2) The binary neighborhood systems of TYPE-attribute: a) VCN Chinese = { Chinese, Vietnamese } b) CN Chinese = { Chinese, Vietnamese, Japanese } c) FN Chinese = { Chinese, Vietnamese, Japanese, American } d) VFN Chinese = { Chinese, Vietnamese, Japanese, American, French } e)... f)... 3) The binary neighborhood systems of PRICE-attribute: a) VCN moderate = { moderate, expensive } b) CN moderate = { moderate, expensive, inexpensive } c)... d)... Now, let us consider the following query: Q 1 : Select a Restaurant which is Japanese, moderate in price and located in Westwood. The traditional database will return a null answer. Then the user may vary his condition slightly and issue another query until either the user is too tired to proceed or he finds his answers. For our proposed system, it will supply an approximate answer for the query Q 1, namely, the restaurant Great Wall because the three neighborhood relations indicated that 1) Chinese and Japanese are very close (in TYPE) 2) St Monica and Westwood are very close (in LOCA- TION) 3) Moderate (in PRICE) If the user is willing to extend his neighborhood to a bigger neighborhood, say close-neighborhood, then two additional restaurants Kiku and South Sea will be included in the approximate answers. IV. CLASSIFICATION AND GRANULAR COMPUTING The basic idea here is to apply Bi-neighborhood granular data model to a core technology of data mining, called classification. The fundamental idea is essentially in Viveros thesis [24] under my supervision. It was summarized in [12] A. Neighborhoods System in Data Mining The goal was to build a system that can automatically classify research articles in computer science: Let V be a list of the titles of research articles, and U be a set of keywords. The goal is to classify V in terms of set of keywords. To each article p V, we associate a subset of key words B p, called basic granule. This association is learned from training data. In the terminology of this paper, it is the model in Definition 3 Note that these basic granules may have non-empty intersections, However, it does induce a partition on V, its equivalence class is called centers [13], later we have called it or center set. Two objects are considered equivalent, if they have the same neighborhood (in data space). This equivalence defines a partition on the object space V Each equivalence class is called the center set (of the same basic granules). This is the desired classification for the objects - research articles. Each basic granule (a subset of key words) is supposed to characterize a subfield of computer science. We assign the subfield name as the name of the neighborhood. Let us enumerate the objects in V, namely, the titles of research articles: 1) L 1 = Finding Reduct for Very Large Databases 2) L 2 = Design Optimization of Rough-Fuzzy Controllers Using a Genetic Algorithms 3) L 3 = Graded Rough Set Approximations Based on Nested Neighborhood Systems. 4) L 4 = Belief Function Based on Multivalued Random Variables, 5) L 5 =... Next let us enumerate the subsets of keywords: 1) K 1 = {reduct, rough set, data, relation } 2) K 2 = {control, genetic algorithm, rough set, neural network, fuzzy logic } 3) K 3 = {rough set, fuzzy set, approximation, neighborhood } 4) K 4 = {belief function, random variables, probability, rough set } 5) K 5 =... Table II illustrates the idea of classification. The first column is the list of papers to be classified. The second column is the set of lists of key words or phrases used to classify the articles. The third column is the name of each class or subfields. 4

4 Object Basic Basic Space Neighborhood Concept V KeyWord Subfield Space U Name L 1 K 1 rough set theory L 2 K 2 intelligent control L 3 K 3 theory of neighborhood systems L 4 K 4 theory of evidence L 5 K 5... TABLE II ABINARY RELATION ON TWO UNIVERSES Table K TOG(K) V F G F -granule G-granule e 1 30 foo {e 1,e 2,e 6,e 7 } {e 1,e 4,e 6 } e 2 30 bar {e 1,e 2,e 6,e 7 } {e 2,e 7 } e 3 40 baz {e 3,e 5,e 8 } {e 3,e 5,e 8 } e 4 50 foo {e 4 } {e 1,e 4,e 6 } e 5 40 baz {e 3,e 5,e 8 } {e 3,e 5,e 8 } e 6 30 foo {e 1,e 2,e 6,e 7 } {e 1,e 4,e 6 } e 7 30 bar {e 1,e 2,e 6,e 7 } {e 2,e 7 } e 8 40 baz {e 3,e 5,e 8 } {e 3,e 5,e 8 } TABLE IV TABLES OF WORDS, AND GRANULES: V. ASSOCIATION RULES AND GRANULAR COMPUTING The main theme in this section is to show how one can represent a relational table by granules. In other words, data processing and data mining are reduced to granular computing. The advantages of such reductions are plenty, for example, one then can reduce association rules mining to a problem of solving linear inequalities [18] and determine all possible features [17]. A. Relational Table and Granular Data Model The following illustration essentially adopt from [3]. In Table III, the first attributes, F, would have three bit-vectors. The first, for value 30, is , because the first, second, sixth, and seventh tuple have F =30. The second, for value 40, is , because the third, fifth, and eighth tuple have F =40; Next, we note that a bit-vector can be interpreted as a subset, called granule, of V. For example, the bit vector, , of F = 30 represents the subset {e 1,e 2,e 6,e 7 }, similarly, , of F = 40 represents the subset {e 3,e 5,e 8 }.This explained how Table III is transformed into Table IV. V F G e 1 30 foo e 2 30 bar e 3 40 baz e 4 50 foo e 5 40 bar e 6 30 foo e 7 30 bar e 8 40 baz TABLE III ARELATIONAL TABLE K Note that the collection of F -granules forms a partition, and hence induces an equivalence relation, Q F ; similarly, we have Q G. Pawlak called the the pair (V,{Q F,Q G }) knowledge base. Since knowledge base often means something else, so, instead, we have called it granular structure or granular data model(gdm)in previous occasions. Based on such representation, we can show that association mining can be transform into solving linear inequalities; see [18] VI. CLUSTERING AND GRANULAR COMPUTING A. Concept Based Document Clustering In this section, we explain the applications of granular computing to clustering, one of the core technology in data mining. Internet is an information ocean, we say it is an ocean, because traditionally database searches are often referred to navigations. With such large amount of information, clustering or classifying semantically a set of documents has been a contemporary challenge. There are many proposals. A fairly popular approaches are clustering by Latent Semantic Indexes(LSI): Though it has named semantic index, these numerical vectors are actually statistical data of keywords; It is unclear where the semantics of documents entering the data. In a series of papers, we have presented a novel approach that obviously has captured the concepts of the contents in a set of documents. Here is a few important points that summarized our approach 1) A token is a document is regarded as a keyword, if its tf-idf values is high. 2) A set of keywords is considered as keywordassociation(frequent co-occurrence set of keywords), if the HIGH DIMENSIONAL tf-idf value is high. 3) The totality of such selected keywords(vertexes), and keyword-associations (simplexes) determines a simplicial complex, called Granulated Latent Semantic Space, which is topologically equivalent to a triangulation (granulation) of a polyhedron. 4) So we have introduced the Euclidean topology into the semantic space of the documents without using a metric. Based on this topology, we propose the following conceptual structure: 1) A PRIMITIVE CONCEPT is represented by a simplex of maximal dimension [20]. 2) A CONCEPT is represented by a connected component. 3) An IDEA is the whole polyhedron. Then we can use these concepts to cluster the documents. Example 3: This example is taken from TAAI 2005 [?]. The universe S 3 consists of twelve keywords that are extracted from a set of documents by some index such as tf-idf. For clarity, we will represent these keywords by alphabets: Data=a, Task=b, Base=c, Group=d, Language=e, Programming=f, Dependency=g Logic(al)=h, Implement=w Function=x, 5

5 Diagram=y, Specification=z. In Figure 1, we have a simplicial complex that consist of twelve vertices that are organized in the forms of 3-complex, denoted by S 3. 8) S(w, x, y, z) =(w, x, y, z) is S(Implement, Function(al), Diagram, Specification) From syntax point of view the first PRIMITIVE CONCEPT 1, that is Data Base Task Group, and the last PRIMITIVE CONCEPT 1 that is Implement Function(al) Diagram, Specification are unrelated. However from Semantic view they are related, since there is a sequence of primitive concepts relate these two concepts together. VII. LEARNING AND GRANULAR COMPUTING Fig. 1. A complex with twelve vertexes. 1) The 3-simplex S(a, b, c, d) (granule of size 4) a) Its four 2-faces (sub-simplexes) S(a, b, c), S(a, b, d), S(a, c, d), S(b, c, d), and b) Its six 1-faces S(a, b), S(a, c), S(a, d), S(b, c), S(b, d), S(c, d), c) Its four 0-faces (vertices), a, b, c, d. 2) The 3-simplex S(w, x, y, z) (granule of size 4) a) Its four 2-faces S(w, x, y), S(w, x, z), S(w, y, z), and S(x, y, z), and b) Its six 1-faces S(w,x), S(w,y), S(w,z), S(x, y), S(x, z), S(y, z) c) Its four 0-faces (vertices), w, x, c, z. 3) The 2-simplexes (granule of size 3) that are lying between two 3-simplexes: S(a, c, h), S(c, h, e), S(e, h, f), S(e, f, x), S(f,g,x), S(g, x, y) and 4) Some of their 1-faces (granule of size 2): S(a, h), S(c, h), S(c, e), S(h, e), S(e, f), S(h, f), S(e, x) S(f,x), S(f,g), S(g, x), S(g, y); non of them are maximal. 5) Their 0-simplex faces (granule of size 1;vertex) a, b, c, d, e, f, g, h, w, x, y, z The simplicial complexes is regarded as a knowledge base of the documents. We will regard the simplexes (closed association rules) of maximal dimensions as PRIMITIVE CONCEPTs: PRIMITIVE CONCEPT 1 is 3-simplex S(a, b, c, d) PRIMITIVE CONCEPT 2 is 2-simplex S(a, c, h) PRIMITIVE CONCEPT 3 is 2-simplex S(c, h, e) PRIMITIVE CONCEPT 4 is 2-simplex S(e, h, f) PRIMITIVE CONCEPT 5 is 2-simplex S(e, f, x) PRIMITIVE CONCEPT 6 is 2-simplex S(f,g,x) PRIMITIVE CONCEPT 7 is 2-simplex S(g, x, y) PRIMITIVE CONCEPT 8 is 3-simplex S(w, x, y, z) Let us reexpress them in terms of keywords 1) S(a, c, b, d) is S(Data, Base, Task, Group) 2) S(a, c, h) =S(h, a, c) is S(Logic(al), Data, BASE) 3) S(c, h, e) =S(h, c, e) is S(Logic, Base(d), Language) 4) S(e, h, f) =S(h, f, e) is S(Logic, Programming, Language) 5) S(e, f, x) =S(x, f, e) is S(Function, Programming, Language) 6) S(f,g,x) =S(f,x,g) is S(Programming, Function(al), Dependency) 7) S(g, x, y) =S(x, g, y) is S(Function(al), Dependency, Diagram) What is learning? For numerical learning, we will take a very simple view as follows: Learning is extending a partially defined numerical function on a finite number of points to a fully defined function. In such learning, training set and background knowledge are often used. Background knowledge can be probability distributions, a family of basic knowledge (classical sets/generalized sets/functions) and etc. This is vast area, and hidden many interests idea [?]. In this paper, we just stay in touch, more details will be reported in future. A. Numerical Learning via Function Granules Next we will explain the universal approximation theorem of neural networks, as well as fuzzy systems, in term of granular computing. Here are our observations: [11] Learning in neural networks is a methodology of determining proper linear combination of activation functions that interpolates a function that is partially defined on a finite number of points. A learned or trained neural network is a mathematical model that approximates a target function by a linear combination of activation functions The input to a neural network is often represented as a training set of data. This set of data defines the function only on a finite number of points. So neural networks is a methodology of interpolating the given training data by a linear combination of activation functions. If one uses different activation function, one will get different interpolation of input data. Intuitively, one can imagine activation functions as a given set of small rules of all shapes. The learning is to use these rules (background knowledge) to draw a curve passing through finite points(training data). There are several well known neural networks that are universal approximates; we regards those activations are basic granules. In fuzzy system, the background knowledge is in the library of membership functions [5] VIII. ACCESS CONTROL IN INTERNET This is a revision of the exposition in Granular Computing I [19]. We will view access list as a basic knowledge about the object. So to manage a massive list of access control (in internet security) becomes a knowledge engineering problem; the details will be in the future report. 6

6 IX. CONCLUSIONS -AIENGINEERING We will refer to manage massive data mining, learning, knowledge engineering, and etc. as AI engineering. We believe granular computing is the fundamental infrastructure for such activities. In this paper, we have superficially addressed issues related to Granular computing in many AI related areas: Intelligent Control in SectionI-A, Data Mining in Section IV, V, VI, Learning in Section VII, Uncertainty in Section III, and even Computer Security in Section VIII. I hope we have convinced readers that granular computing is an infrastructure that may supports many concepts and processing. In this paper we have focused more on AI related areas. Next let us discuss some new integration technologies or concepts that can/should be addressed in granular computing. A granular data model has two components U and β. Most of the focus has been towards understanding U. Now will focus on β. We will illustrate by mathematics some possible issues that are hidden in integrating subtasks and quotient tasks derived from β. Let us consider a set theoretical view first: Assume we have a partition, β, of the integers Z (in the category of sets) 1) The partition β has two sets (equivalence classes); both are equivalent to Z as sets. a) [0] 2 = {..., 2, 0, 2, 4,...}( Z as sets). b) [1] 2 = {..., 3, 1, 1, 3,...}( Z as zets). 2) The partition has a quotient set Z 2 ={[0] 2, [1] 2 }. Now, if we are given two copies of Z that are set theoretically isomorphic to [0] 2 and [1] 2, but has not knowledge about how they should fit to each other to form the original Z. In other words, if two subtasks [0] 2 and [1] 2 of Z and a quotient task Z 2 ={[0] 2, [1] 2 } are given, could we find the Z back and are such solutions unique? As set theory both answers are yes. Next, let us consider the situation where β is more structure than a set. We will add a simple structure into the integers Z, namely, the addition. Such mathematics is called commutative group or Abelian group. Let us assume we have 1) The partition β is generated by a subgroup, namely, the subgrouo of even integers. So the two equivalence classes are a subgroup and a coset; a) [0] 2 = {..., 2, 0, 2, 4,...}( Z as a subgroup). b) [1] 2 = {..., 3, 1, 1, 3,...}( Z as a set). 2) The partition forms a quotient group Z 2 ={[0] 2, [1] 2 }. Now, if we are given a subgroup [0] 2 (this will generated two equivalence classes), and a quotient group Z 2 ={[0] 2, [1] 2 }, could we find Z (as group) back? There are two solutions: one is Z Z 2 and the other is Z. This is simple math, so we have the solution. In real world problem, [0] 2 represents subtasks and Z 2 represents quotient tasks. The latter on is high level problem that is resulted from information hiding. In many AI problem this level of integration has never been discussed - this is new AI-Engineering problem. 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