Granular Computing:

The Concept of Granulation and Its Formal Theory I

Tsau Young (T. Y.) LinDepartment of Computer Science, San Jose State University

San Jose, California 95192, USAtylin@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 earthare 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 ideal-ized/simplified granulation into partitions (=equivalence relations), and have developed thepartition into a fundamental concept in mathematics, for example, congruence in Euclideangeometry, quotient structures (groups, rings, etc) in algebra, the concept of ”a. e.” (almostevery where) in analysis. Nevertheless, the notion of partitions, which absolutely does notpermit 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 beingsthat 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 ofthe idea. Lately, however, the concept seems to have reached its steady state. The goal of thispaper 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 academicyear 1996-97, when I arrived at UC-Berkeley for my sabbatical leave, Zadeh suggested granularmathematics (GrM) to be the area of research. To narrow down the scope, I proposed theterm granular computing to label the area [42]. Therefore, at the very beginning, GrC is thecomputable part of the granular mathematics.

What is Granular Mathematics (GrM)? In his 1979 paper [40], Zadeh already had implicitlyexplained 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 innature.”

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Our views are the same, but I adopted the incremental approach: By mapping Zadeh’sintuitive definition [41] to Neighborhood System(NS), I took NS as the GrM, and regarded itas the first GrC model [17], [18], [19].

What is Neighborhood Systems(NS)? Totally from differnt context, namely, approximate re-trival, in 1988-89, I generalized the notion of tolological neighborhood systems (TNS) to thatof Neighborhood Systems(NS) by simply dropping the axioms of topology [9], [11]. In thisview, each neighborhood is a unit of uncertainty; see Section 5. But in later part of the sameyear (1989), when I considered the Chinise wall computer security policy model (CWSP), aneighborhood was regarded as a unit of information (known basic knowledge) [10]. In otherwords, 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 steadystate. 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’sintuition, 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, theformal definition of granulation, is presented. In this section, we also include Zadeh’s informaldefinition. 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. Precislyspeaking, this view needs some stipulation; it actually implies implicitly that we are ignoringthe interactions among objects. Recall that in set theory, every point is discrete in the sensethere are no interactions among elements. However in GrC, one of the manin concerns is theinteractions among grnanules, so the collection of granules, called granular structure (GrS), ismore than a set of granules. In fact, there are three states for GrS; we will discuss this in thenext paper.

For now, to understand some of the delicate nature of GrC, let us recall some practices inmathematics.

Example 1 Examples from Algebraic geometry: Let Z be the ring of integers. That is, we areconsidering 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 ZLet p be a prime number, then the prime ideal is a subset

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{. . . ,−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 bySpec(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 = {e0, e1, e2, e3} be a finite set.

1) from Partition Theory: Let β be the collection {{e0, e1}, {e2, e3}} of subsets. As all subsetsare mutually disjoints, β is an honest classical set.

Next, we consider a very common example, which often has not been carefully considered. Letβ be the collection of all subsets of U . In this case, there are interactions among these subsetsas 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 ”inter-section” and ”union.”

4) In Lattice Theory: β is a lattice, if we do consider the ”union”, ”intersection” and ”inclu-son” 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 ={e0, e1, e2, e3} to be a set of linearly indepdent points in a Euclidean space. then β can beinterpreted as a simplicial complex as follows:

1. U = {e0, e1, e2, e3} is a collection of vertices

2. β is a collection of simplexes

(a) Four singletons are the four 0-simplexes: ∆i = {ei}, i = 0, 1, 2, 3‘.

(b) Six subsets of two elements are the six 1-simplexes: ∆ij = {ei, ej}, i < j = 0, 1, 2, 3.

(c) Four subsets of three elements are the four 2-simplexes: ∆i = {ei, ej, ek}, 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.

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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 isgranulated 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 writedown a membership function to represent a granule, such as head, neck or body. However, eachexpert may come up a distinct membership function, and therefore may have a distinct theoryof GrC on human body. This is not a satisfactory theory. As there is no unfied view on all ofthese 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. Ofcourse, 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 notionled to the invention of calculus by Newton and Leibniz. Actually the idea was much moreancient; it was in the mind of Archimedes, Zeno, and etc. Yet the solutions were in moderntime. It led to the theory of limit (18th century), topology (early 20 century [30]) and non-standard 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 andSecond GrC Model; they have been referred to as neighborhood systems and partial coveringsrespectively in pre-GrC time.

E3 Classical Case: Partition (equivalence relation)

This class of examples have been well studied in mathematics and and recently in computerscience by rough set community.

E4 Granules of Uncertainty from Quantum Mechanics

Heisenberg uncertainty principle states that, in general, neither the momentum nor the positionof 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 positionsand vice versa.

E5 Granules of Knowledge from Computer Security

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In many computer systems, . Discretionary Access Control model assigns each user a set ofusers(friends) who can access his files. However, we also consider a set of users(foe), who cannotaccess 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 GrCModel. It was called binary neighborhood system in pre-GrC terminology. Mathematically, itis equivalent to a binary relation. Geometrically, a binary relation is a graph or network. Theyare the major data structures in computer science. The example [E4] is also in the category ofrandom 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 cov-erings 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 hastwo 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 beviewed 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 wasobserved by Z.Pawlak in late 1982 and Tony Lee in early 1983 [7], [32]. An attributevalue can be regarded as the name of an equivalence class. In this view, a frequentitemset is an intersection of equivalence classes, whose cardinal number is large than agiven 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 regardedas an abstract ordered simplex. Moreover the apriori principle of frequent item sets turnsout to be the closed condition of simplicial complex [?]. Recall that a simplicial complex (ofkeywords) consists of two objects: one is a finite set of vertices (keywords). Another one isa family of ordered/oriented/unordered subsets, called ordered/oriented/unordered simplexesthat satisfy the closed condition, namely, any subset of a simplex is a simplex. This is animportant mathematical structure in algebraic/combinatorial topology. Currently, it is findingits way to web technology [23], [?], [25].

E8 Granules of Knowledge from Social Networks.

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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 cannotexchange their roles freely. We can view the collection of roles as a relational schema. If wedo so, then a committee is a tuple. Observe that different types of committees have differentschema. The set of committees under the same schema forms a relation. A collection of n-naryrelations 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 tomore complicated objects.

First, we observe that all examples can be fuzzified (type I fuzzy set), which is fully charac-terized by a membership function. Mathematically, they are bounded functions. Moroever, theexample [E4] in Heisenbeerg Uncertainty Principle is actually a granulation of random varibles(measurable functions). Hence, it should be nature to extend the consideration of membershipfunctions and measurable functions to general functions. So we include the following exampleinto 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 universalapproximation peoperty will be regarded as functional granulation. In fact, we will extend itto 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, how-

ever, ”how to compute” is an integral part of computing. So GrC includes some mathemati-cal/computational practices.

E11 A granule can be a subprogram in a program, or a lemma in a mathematical proof, whenthe 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 Thisidea is modeled in 7th GrC Model.

A lemma in a mathematical proof is a granule conceptually, however, we should not includeit in GrC unless the mathematical proof itself is computable. However, in this class, we mayextend the idea of 7th GrC Model to GrM..

Let us sumarize this section by Zadeh’s Intuitive Definintion.

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• ”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 indistin-guishability, 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” someconstraints 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 simplermodels 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 afamily of subsets. Then the pair (U, β), is called the 2nd GrC Model. If we use pre-GrCterminology, β should be called a Partial Covering(PCov). However, most of those papersadressed 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 exertedon 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 relationaldatabase). In this case the collection of granules are tuples of some relations. These are modeledin 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 indexset. As this paper will include GrM, we will take the following convention: the lowercase letter, say k, denotes the parameter that runs though a set, K, whose name is thecorresponding cap letter.

To make the understanding of the 5th GrC model easier, we explain the generalizationprocess as follows:

1. U is generalized to U = {Uhj | h ∈ H; j ∈ J}

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2. . . . . . . . . .

3. F k ⊆ U is generalized to

Rk ⊆ ∏{Uklk

| K ⊆b H; Lk}

4. . . . . . . . . .

2) 5th GrC Model

1. Let U = {Uhjh

, h ∈ H, jh ∈ Jh} be a given family of classical sets, called the universe.Note that distinct indices do not imply the sets are distinct.

2. Let∏{Uk

lk| K ⊆b H; Lk ⊆b Jk} be a family of Cartesian products of various

lengths n, where n = |Lk| is the cardinal number of Lk that may be infinite. Here”⊆b” means a sub-bag. Recall that a bag is a set that allows the repetition of someelements [4].

3. Recall that an n-ary relation is a subset Rm of a product space in the previous item.

4. Let β = {Rm | m ∈ M} be a given family of n-ary relations given in previous itemfor various n; note that n can be infinite.

Then the pair (U , β), called Relational GrC Model, is the formal definition of Fifth GrCModel

4.2 Category Theory Based Models

First, we would like to observe that 8th GrC Model is abstractly the same as the category ofrelational databases [13]. In other words, from the point of views of mathmatical structuresthe 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 domainX 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 ofmorphism f : X −→ Y and g : Y −→ Z their composite g ◦ f : X −→ Z Thesesatisfy 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

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(b) Identity. For every object Y there is a morphism IY : Y −→ Y such that iff : X −→ Y , then IY ◦ f = f and if h : Y −→ Z, then h ◦ IY = 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 themaps.

3. The category of crisp/fuzzy sets with crisp/fuzzy binary relations as morhisms: Theobjects 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 UX is the power set P (X) of a classical set X.Let UY be another object, where Y is another classical set. The morphisms are the maps,P (f) : UX −→ UY that are induced by maps f : X −→ Y .

5. The category of topological spaces: The objects are classical topological spaces. Themorphisms are the continuous maps.

6. The category of neighborhood system spaces (NS-space): The objects are NS-spaces; seeFirst 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 = {Chj | h ∈ H, jh ∈ Jh} be a family of objects in the Category CAT.

2. There are families (which are bags) of Cartesian products∏{Ck

lk| K ⊆b H; Lk ⊆b

Jk} of various lengths n, where n = |Lk| is the cardinal number of Lk that may beinfinite. They are called product objects.

3. An n-ary relation object Rj 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 modelof granulation.

By specifying the general category to various special cases, we have all the models. We willexplain 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

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(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, wehave 4th GrC Model.

(b) By limiting the number of binary relations in 4th Grc Model to be one, we have 3rdGrC 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 5

4.3 Overview of GrC Models

Schematically we summarize the relationships, based on the Granular Structures, of early GrCModels 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.

8th GM ⇒

5th 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 intwo ways:

1. The formal infinitesimal granule in non standard world (NSW).

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2. Topological Neighborhood System (TNS) in standard world.

We will focus on TNS. It is important to observe that the ancient intuition of infinitesimalgranules (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 theaxioms 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 assignmentthat associates each point p a family of subsets, TNS(p), that satisfies the following axiomsof topology:

(a) If N ∈ TNS(p), then p ∈ N ,

(b) If N1 and N2 are member of TNS(p), then N1 ∩ N2 ∈ TNS(p)

(c) If N1 ∈ TNS(p) and N1 ⊆ N2, then N2 ∈ TNS(p),

(d) If N1 ∈ TNS(p) then there is a member N2 ∈ TNS(p) such that N2 ⊆ N1, andN2 ∈ TNS(q) for each q ∈ N2(that is, N2 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. 2Y is the family of all crisp subsetsof Y , where Y = P (U). In other words, NS associates each point p in V , a family NS(p) ofcrisp/fuzzy subsets of U . Such a subset is called a neighborhood (granule) at p, and NS(p) iscalled a neighborhood system at p.

Definition 4 The 3-tuple (V, U, β) is called First GrC Model (Local GrC Model), where β isa neighborhood system (NS). If V = U , the 3-tuple is reduced to a pair (U, β). In addition, ifwe 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 = {Bi | i ∈ I} is a collection ofbinary relations Bi on V × U . Let Bi(p) = {(p, x) | x ∈ U} and p ∈ V . Then NS(p) ={Bi(p) | ı ∈ I} defines a neighborhood sysgtem on p. Let p vary through V , we have an NSand (V, U,NS) is a 1st GrC Model.

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5.1 The Concept of ”Near” and Contexts

The following arguments are adopted from my pre-GrC paper [16] The notion of near is ratherdifficult to formalize. Let us examine the following examples.

1. Is Santa Monica ”near” Los Angels? Answers could vary. For local residents, who havecars, 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 thetolerance radius, in other words, in the given context.

Intrinsically ”near” is a subjective judgment. One might wonder whether there is a scientifictheory 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 selectedcontext)

Is 1.73 ”near”√

3 ?

With this context selection, that is, 1/100 is acceptable error, then 1.73 is near√

3 ! On theother 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 allpositive real numbers. We often use ε to denote the variable that varies through such a contextset.

Similarly, let us assume a neighborhood system has been assigned to each city in Los Angelesarea: For example, based on car driving, public transportation, walking and etc, we assign aneighborhood to each city for each context. Under such a concept of neighborhood system, thequestion 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 possiblecontexts.

2. Under this model, in an application, selecting a context means selecting a fixed neighbor-hood 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”, thenSanta Monica is ”near” Los Angels.

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Example 4 Let the collection of ε-neighborhoods at each point be the neighborhood system ofthe real numbers R; Then(R, ε-neighborhoods) provides the proper contexts for discussing ”near”answers( approximate answers), where ε could take any real value. (R, ε-neighborhoods) is aFirst GrC Model [?]

Now, we re-state the previous example using this First GrC Model

1. Assuming we have agreed ε = 1/100 is acceptable, then 1.73 is ”near”√

3

2. But, if we have only agreed ε = 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 ε > 0 (a context at zero), there is a number N= [1/ε] + 1, such that, ε > 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 (ε, δ)-definition of limit can spot the origin of neigh-borhood systems. Such a context free (all possible contexts) answer is precisely the classicalnotion of limits, limn→∞1/n = 0. Using our language, we may say that limit is the context freeanswers of ”near”

Perhaps we should also point out here that there is no context free answers for the questionwhether two points are ”near.”

• A Lesson

1. Each granule provides a context, a state or an agreement as what to be consideredas ”near,”

2. Granulation, namely beta, provides the complete contexts/states that can be usedin reasoning about ”near-ness”.

Brief pre-GrC historical notes:

1. In 1988-89, Lin generalized TNS to the Neighborhood Systems(NS) by simply droppingthe (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 relationson 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].

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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. Theterm 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 NSas 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 , iscalled 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-collectionof all members of the partial covering β, that contains p, as a neighborhood system at p, thenthis Second GrC model is an example of First GrC model.

The modern example, simplicial complexes, is an important example of such a model: Asimplicial complex consists of a set of vertices and a family of subsets, called simplexes, thatsatisfies 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]. Recallthat [E4] concludes that, a precise measure of the momentum can only determine a (proba-bilistic) ”neighborhood” of positions; and [E5] concludes that in computer security, the Discre-tionary Access Control Model (DAC) assigns to each user p a family of users, Yi, 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 asubset, B(p), of ”basic knowledge” (a set of friends or a ”neighborhood” of positions ).

p −→ B(p) = {Yi, i = 1, . . . } ⊆ U

Such a set B(p) is called a (right) binary neighborhood and the collection {B(p) | ∀p ∈ V } iscalled the binary neighborhood system (BNS).

Definition 6 Third GrC Model: The 3-tuple (U, V, β), where β is a BNS, is called a BinaryGrC Model. If U = V , then the 3-tuple is reduced a pair (U, β).

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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 asingleton 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 geomet-rically as graphs, networks, forest and etc. So Third GrC Model has captured most of themathematical 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, iscalled Multi-Binary GrC Model. This model is most useful in data bases; hence it has beencalled Binary Granular Data Model(BGDM), in the case of equivalence relations, it is calledGranular 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 Fourthmodels 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 ahuman society then a group of people may be drawn into a committee with distinct roles, suchas the chair, vice chair, secretary, treasurer, and etc. As every member has different role, theycan not be swapped around. So the committee is not a set; it is a tuple under the schema thatconsists of distinct roles.

Definition 8 5th GrC Model:

1. Let U = {Uhj , 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.

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2. Let U j1 × U j

2 × . . . be a family of Cartesian products of various length.

3. Recall that an n-ary relation is a subset Rj ⊆ U j1 × U j

2 × . . . U jn.

4. Let β = {R1, R2, . . . } 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 FirstOrder 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 maynote 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 evengeneralized functions.

Fuzzy sets are described by membership functions, so granules can be regarded as membershipfunctions; note the First to 5th GrC Models include fuzzy sets. Hence, we consider furthergeneralizations: granules are functions, random variables (measurable functions) generalizedfunctions (e.g. Dirac delta functions) .

In the case, a granule is a function, we may require that the granular structure (the collectionof granules) has the universal approximation property, namely, any function in the universe canbe approximated by the functions in the collections. The membership functions selected in fuzzycontrols do have such properties. In neural networks, the functions generated by the activationfunctions also have such property [31]

In the case of probability/measure theory, quantum mechanics may be a good guidingexample.

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 subpro-grams in a computer program, or a computer or cluster of computers in grid/cloud computingare 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 printingform 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 watchfor new development.

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9 Qualitative Fuzzy Sets - A Proposal

Fuzzy theories have been driven by applications. Naturally implicitly or explicitly, some con-texts of their specific applications may have imposed into their theoretical frameworks. In thispaper, 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 membershipfunctions, each of which represents a state of the ”real world fuzzy set” (in different time orsituation). 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 ofthe ”real world fuzzy set”

3. The granule .provides the complete states/contexts that can be used in reasoningabout ”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 ”realworld fuzzy set.”

We suggest the readers to detour to Section 11 for a quick glance on the various notions ofsofsets (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 worldfuzzy set.” In other words, selecting a membership function(similar to selecting an ε) isequivalent 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 operationcirc defined below (Section 9.1) .

(b) Each neighborhood N ∈ β, is a collection of membership functions that representsa 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 gen-erated by unions and intersections of C. Then a member C of β (a granule) represents a ”realworld fuzzy set.” The granule C is a called a qualitative fuzzy set; each membership function inC represent a state of the ”real world fuzzy set.”

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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 FirstGrC Model. The key question is: Can we generate a Ninth GrC Model from the NS? Here isthe required constructions:

Let NS(p) be the neighborhood system at p of a First GrC Model; Let N(p) represent anarbitrary neighborhood of NS(p) Let CN(p), called the center set of N(p), consists of all thosepoints that have N(p) as its neighborhood. (Note CN(p) is called center set; in the case oftopological 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 intersec-tions of all neighborhoods in NS(p). Then a hard question is: Does the intersection of twoneighborhoods (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 CN(p) ∩ CN(q) 6= ∅.

2. N(p) ∩ N(q) is not in any G(p) ∀ p, iff CN(p) ∩ CN(q) = ∅.

In GrC, we regard N(p) as a known basic knowledge, and we defined the knowledge opera-tions [?]: Let ◦ be the ”and” of basic knowledge. For technical reasons, the ∅ is regard as apiece of the given basic knowledge.

Definition 13 New operations

1. N(p) ◦ N(q) = N(p) ∩ N(q), iff CN(p) ∩ CN(q) 6= ∅.

2. N(p) ◦ N(q) = ∅, iff CN(p) ∩ CN(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 CB(p) = CB(q).

2. B(p) ◦ B(q) = ∅, iff CB(p) ∩ CB(q) = ∅. Note that B(p) ∩ B(q) may not empty, but it isnot a neighborhood of any point.

Observe that in Binary GrC Model, two basic knowledge are either the same or the set theo-retical intersection does not represent any basic known knowledge.

Theorem 1 Given a First GrC Model (U,NS), then the Boolean algebra β generated from NSusing ◦ 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 theBoolean algebra generated by C using unions and intersections.

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10 Conclusions

In Springer Encyclopedia, we have proposed a category theory based model, as the FormalModel for GrC. We said ”the model can be specialized into various category to realize all theclassical examples, including the first example, the granulation of human body. However, theclaim on the realization of the first example was not in printing form This paper is to realizethis claim, and we have shown that

1. The universe of discourse is the domain of interests. In this paper, the domain of in-terests 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 BooleanAlgebraic Structure on the collection of granules. Note that the ”intersections’ are actuallythe knowledge operation ◦.

3. A granule consists of a collection of membership functions is regarded as a representationof an elastic membership function of a real world fuzzy set. Elasticicity is dynamic, so torepresent a dyanmic system, we have to use a trajectoiry of states to represent. So thegranule is the trajectory conists of all the ”current” states.

4. Each membership function in the granule represents a state of this elastic membershipfunction

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 toa 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 elements0, 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 fuzzysets. From now on M will be the unit interval [0, 1]. The collection of all FX’s is called themembership function space on U, denoted by MF(U).

As in [15], let Col be a neighborhood system on a membership functions space. Eachneighborhood represents a qualitative fuzzy set; we shall call it sofset (- not a typo)

Definition 16 A member of Col is a:

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1. W-sofset (Weighted Soft Set), if Col consists of singletons. A membership function istreated as a characteristic function of a soft set. This is essentially Type I fuzzy set.

2. F-Sofset (Finite-multi Soft Set), if Col consists of finite sets only.

3. P-Sofset (Partitioned Soft Set), if Col forms a crispy partition. This is mathematicallya most beautiful theory. Realizing that a fuzzy set (that tolerates perturbation) has to berepresented by a set of membership functions. Each set represents one and only one fuzzyset. Then the space of membership functions is partitioned into equivalence classes. SoP-sofset theory is very elegant and beautiful. However, one may wander how could therebe a natural partition in a ”continuous” membership function space. This lead us to amore general point of view, namely, the next few items.

4. B-Sofset (Binary Neighborhood Soft Set), if Col forms a binary neighborhood system (i.e.,the neighborhood system is defined by a binary relation). Binary neighborhoods are geomet-ric view of binary relation. Intuitively, related membership functions are ”geometrically”near to each other. A binary neighborhood system is equivalence to an abstract binaryrelation,

5. C-Sofset(Covering Soft Set), if Col forms a covering.

6. G-Sofset(G-covering Soft Set), if the covering Col forms a semi-group under intersection(G may contain empty set)

7. N-Sofset (Neighborhood Soft Set), if Col forms a neighborhood system (NS); see Sec-tion ??.. This is the target concept, qualitative fuzzy set. It is most general case; itcontains all previous cases.

12 Future Directions

Granular Computing is still in its inception stage; possible directions are wide open. Here wewill 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 bespecialized 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 firstexample is not in printing form yet. However, this author feels that it is important to informthe readers that is occurring.

The key to realize the first example is based on the category of qualitative fuzzy sets orsofsets(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

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Given a granular structure, we associate it with four structures (including itself). Among themquotient and knowledge structures are mathematical consequences of granular structure (if itis 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. Weurge the readers to read Zadeh’s article.

3) Imported Concepts

For information integration (this may correspond to Zadeh’s term, ”organization”), we haveillustrated the idea imported from homological algebra. It is unclear if we have imported thecorrect 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 thebase of uncertainty, while GrC regard a granule as a unit of uncertainty (such as uncertainty inquantum mechanics) Also the approximation theories are different. Of course, there are otherdifferences; we skip.

5) GrC, Databases and Data Mining

As we have pointed out that the categorical structures of databases and GrC are similar; atthe same time, we need to point out the differences in semantics. Nevertheless, we are lookingforward to the transfer of database technology to GrC. For data mining, please see databasesection on the articles by this author on ”deductive data mining using GrC,” and ”miningdecision rules using RST.”

6) GrC and Fuzzy Logic

Most of expositions have been based on classical sets (and fuzzifiable concepts) For more in-trinsic 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 havemany interactions with wide variety of areas.

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