{power-supply, amplifier, output-unit}. Thus, the attribute function [f] is represented by the functional parameters of these components, for example, the voltage of power supply, the amplification of amplifier, etc. How to choose these parameters in order to have more information regarding the fault of elements in X can be handled by an experienced troubleshooter easily. A knowledge-driven approach is generally adopted by many troubleshooters.2.3. The Extraction of Information on Coarsely Granular Levels
Solving a problem in space 
means investigating the attribute function of each element and the relationships among elements of X, i.e., the structure T, through analyzing and reasoning on domain X. If the multi-granular computing is adopted, the same attribute function f and structure T will be gained by analyzing and reasoning on a certain coarse granular world [X] of X. Thus, [f] and [T] in space 
must represent f and T to some extent, respectively. In Chapter 2, we have already discussed how [T] is induced from T when T is a topology or a semi-order. The construction of [f] will be discussed in this section.
means investigating the attribute function of each element and the relationships among elements of X, i.e., the structure T, through analyzing and reasoning on domain X. If the multi-granular computing is adopted, the same attribute function f and structure T will be gained by analyzing and reasoning on a certain coarse granular world [X] of X. Thus, [f] and [T] in space must represent f and T to some extent, respectively. In Chapter 2, we have already discussed how [T] is induced from T when T is a topology or a semi-order. The construction of [f] will be discussed in this section.Generally, constructing [f] from f is a domain-dependent problem. For example, when diagnosing an electronic instrument, our aim is to find the faulty element (or elements) of the instrument. The domain X is all electronic elements in the instrument. The attribute function f(x) is the functional parameters of each element in X. The aim of the diagnosis is to find which f(x) (or f(x)’s) is abnormal. In the case of the multi-granular diagnosis, the diagnosis first begins from a certain coarse granular level consisting of some components, for example, {power-supply, amplifier, output-unit}. Thus, the attribute function [f] is represented by the functional parameters of these components, for example, the voltage of power supply, the amplification of amplifier, etc. How to choose these parameters in order to have more information regarding the fault of elements in X can be handled by an experienced troubleshooter easily. A knowledge-driven approach is generally adopted by many troubleshooters.
{power-supply, amplifier, output-unit}. Thus, the attribute function [f] is represented by the functional parameters of these components, for example, the voltage of power supply, the amplification of amplifier, etc. How to choose these parameters in order to have more information regarding the fault of elements in X can be handled by an experienced troubleshooter easily. A knowledge-driven approach is generally adopted by many troubleshooters.The construction of [f] still has some general principles. We next discuss these principles.
From the analysis of computational complexity above, we know that the relation between the goal estimation function and additional computation y is 
, where the form of 
affects the computational complexity directly. If constructing a hierarchical structure (or [f]) such that 
, the complexity will be reduced to 
by the multi-granular computing. Conversely, if constructing a hierarchical structure (or [f]) such that 
, generally, only the polynomial complexity can be obtained by using multi-granular computing. At worst, if an improper structure (or [f]) is constructed, the computational complexity may not be reduced.
, where the form of affects the computational complexity directly. If constructing a hierarchical structure (or [f]) such that , the complexity will be reduced to by the multi-granular computing. Conversely, if constructing a hierarchical structure (or [f]) such that , generally, only the polynomial complexity can be obtained by using multi-granular computing. At worst, if an improper structure (or [f]) is constructed, the computational complexity may not be reduced.The preceding discussion implies that the multi-granular computing does not necessarily bring up high efficiency. The key is to construct a proper hierarchical structure. Generally, in multi-granular computing we follow the following principle. When a new problem space 
is constructed from the original one 
, we expect to get from the former as more useful information regarding the latter as possible while keeping the new problem space simple.
is constructed from the original one , we expect to get from the former as more useful information regarding the latter as possible while keeping the new problem space simple.Here, we propose a general principle for constructing quotient space. It is called ‘homomorphism principle’.
Homomorphism Principle
If there is a solution of problem p in space 
, i.e., p is true, then a solution of corresponding problem [p] in 
also exists, i.e., [p] is also true. In other words, if there is no solution of [p] in 
, then there is no solution of p in 
either.
, i.e., p is true, then a solution of corresponding problem [p] in also exists, i.e., [p] is also true. In other words, if there is no solution of [p] in , then there is no solution of p in either.It seems that the homomorphism principle above is used extensively in our everyday life. Some examples are given below.
Example 2.7
Making an annual production plan, one first makes quarterly plans then monthly plans within each quarter and daily plans, etc., hierarchically.
Obviously, if there is an annual plan satisfying some given conditions, there exist quarterly plans, monthly plans and daily plans which satisfy the given conditions.
Conversely, if there is no quarterly plan, monthly plan or daily plan which satisfies the conditions, the annual plan cannot be worked out.
Example 2.8
In a find-path planning of an indoor robot, if the robot environment is rather complicated, it would be difficult to find a collision-free path at once by considering all the details of the whole environment. The path planning might have two levels. The first one, referred to as INTERROOM plan, is the planning of navigation from room to room without considering any details within the rooms. The second level, called INROOM plan, is the level of planning robot movements within the rooms.
It is obvious that if we cannot find a path from room to room, the entire path planning must fail.
We have seen that by the ‘homomorphism principle’, if we fail to find a solution in some parts of a coarse space, the corresponding parts of the original fine space have no solution either and can be pruned off. Clearly, through each granular level computing the search space is narrowed down. We now come to the key issue in our multi-granular computing. Whether the computational complexity can be reduced depends on how much search space is narrowed down through each granular level computing and how much additional computation is needed in order to have such a reduction. This is just the consideration that we need in constructing [f].
2.3.1. Examples
Before going deeply into the approaches to constructing [f], let us now examine some examples.
Example 2.9
In a national economy developing plan, if factories, mines and enterprises are elements of domain X, and 
, x indicates the annual output value of these units. By hierarchical partition, some factories or enterprises compose a department A. The annual output value 
of each department A, certainly, is defined as
, x indicates the annual output value of these units. By hierarchical partition, some factories or enterprises compose a department A. The annual output value of each department A, certainly, is defined as
If 
indicates the productivity of each factory, the productivity for A is defined as
indicates the productivity of each factory, the productivity for A is defined as
Example 2.10
When throwing a ball upwards, 
represents the attributes of the ball's motion at moment 
, where 
is the distance of the ball from the ground, 
is its velocity and 
is a time interval.
represents the attributes of the ball's motion at moment , where is the distance of the ball from the ground, is its velocity and is a time interval.If we are only interested in the qualitative attributes of the motion, i.e., whether the ball goes upwards or downwards, or static, then time interval 
is partitioned into five parts as follows.
.
is partitioned into five parts as follows.
.Let 
where, 
, and 
denotes that the ball is on the ground, 
above the ground, 
goes upwards, 
goes downwards, and 
static. It is easy to have
, and denotes that the ball is on the ground, above the ground, goes upwards, goes downwards, and static. It is easy to have
Note that the range of function 
is defined on some quotient set 
rather than 
. That is, partitioning Y into several equivalence classes and giving each equivalence class a symbolic name, the range of 
is just a set of these names. Sometime, it’s called a qualitative description of 
.
is defined on some quotient set rather than . That is, partitioning Y into several equivalence classes and giving each equivalence class a symbolic name, the range of is just a set of these names. Sometime, it’s called a qualitative description of .In summary, assume that 
is known. 
represents some attribute of element 
. The point is how to choose the value of 
that can properly mirror the feature of the whole set A representing by the attribute 
and satisfies the homomorphism principle above. Generally, 
may be a value from Y or [Y].
is known. represents some attribute of element . The point is how to choose the value of that can properly mirror the feature of the whole set A representing by the attribute and satisfies the homomorphism principle above. Generally, may be a value from Y or [Y].More generally, given some local information of a set, we want to find the global information such that it can depict the features of the whole set. For example, the feature extraction of images in pattern recognition is somewhat similar to the above problem, where 
(color, gray-level) is extracted from each pixel x, while 
represents the feature of regions or the whole image. We next present some common approaches to constructing function 
.
(color, gray-level) is extracted from each pixel x, while represents the feature of regions or the whole image. We next present some common approaches to constructing function .2.3.2. Constructing [f] Under Unstructured Domains
Starting from the domains X where elements are independent or near-independent of each other, we call it unstructured domains.
In terms of statistics, this amounts to ‘mutually independent’ assumption for random variables. In topological terminology, it is equivalent to ‘discrete topology.’ So unstructured is a special structure.
1. 
is a space, where T is a discrete topology and 
, 
is an n-dimensional Euclidean space. Assume that R is an equivalence relation on X. The corresponding quotient space is 
. 
is an attribute function on 
.
is a space, where T is a discrete topology and , is an n-dimensional Euclidean space. Assume that R is an equivalence relation on X. The corresponding quotient space is . is an attribute function on .(1) Statistical method
Since X is unstructured, 
depends on 
completely. From statistics, attribute 
of X can be regarded as a set of mutually independent random variables. Thus, attribute 
of [X] can be defined as some kind of statistics.
depends on completely. From statistics, attribute of X can be regarded as a set of mutually independent random variables. Thus, attribute of [X] can be defined as some kind of statistics.(2) Inclusion principle
If 
, define 
as a specific point within 
where 
is a convex closure of B. For example,
, define as a specific point within where is a convex closure of B. For example,
= the mean of 
on A,
= the center of gravity of 
, etc.In Section 2.3.1, many examples for extracting 
belong to the inclusion principle.
belong to the inclusion principle.(3) Combination principle
If A is a finite set, 
can be defined as follows
can be defined as follows
where g is an arbitrary combination function, for example, 
.2. 
is a set-valued function, i.e., 
, 
is a subset of Y.
is a set-valued function, i.e., , is a subset of Y.(1) Intersection or union method
If Y is a complete semi-order lattice, then 
and 
or 
and 
are equivalent. The method is equal to the inclusion principle in 1.
and or and are equivalent. The method is equal to the inclusion principle in 1.An example is given below.
Assume that A={a set of quadrilaterals on a plane}=
a1={a set of squares}, a2={a set of rectangles},
a3={a set of diamonds}, a4={a set of parallel quadrilaterals},…
Let
={four edges are equal, four angles are equal, the sum of all inner angles is 2π,…},
={two mutually opposite edges are equal, four angles are rectangle, the sum of all inner angles is 2π,…},
={four edges are equal, two mutually opposite angles are equal, the sum of all inner angles is 2π,…},
={two mutually opposite edges are equal, two mutually opposite angles are equal, the sum of all inner angles is 2π,…},….Now, we can use 
to define 
, for example, 
.
to define , for example, .(2) Quotient space method
Thus, 
represents the image of 
on 
via natural projection p. The range of 
is 
.
represents the image of on via natural projection p. The range of is .
In qualitative analysis, 
is partitioned into 
and 
simply, 
denotes as ‘0’ and 
as ‘+’, respectively. R is partitioned into 
, and 
, simply, 
as ‘0’, 
as ‘–’, and 
as ‘+’, respectively. Defining 
, we then have the same results as shown in Example 2.10 of Section 2.3.1. Using the elements (or subsets) in quotient space [Y] to describe [f] is a useful method in qualitative reasoning especially. We’ll discuss the problem in Chapter 4.
is partitioned into and simply, denotes as ‘0’ and as ‘+’, respectively. R is partitioned into , and , simply, as ‘0’, as ‘–’, and as ‘+’, respectively. Defining , we then have the same results as shown in Example 2.10 of Section 2.3.1. Using the elements (or subsets) in quotient space [Y] to describe [f] is a useful method in qualitative reasoning especially. We’ll discuss the problem in Chapter 4.2.3.3. Constructing [f] Under Structured Domains
Under the structured domains, the values of 
that correspond to different element x are interdependent and are related to structure T. So from structure T, we can have some knowledge about 
. Generally, structure T may be classified into three main classes below.
that correspond to different element x are interdependent and are related to structure T. So from structure T, we can have some knowledge about . Generally, structure T may be classified into three main classes below.(1) T is a topologic structure. For example, X is an n-dimensional Euclidean space. Then the Euclidean distance is its topologic structure T. Generally speaking, any metric space is an instantiation of topologic structures.
(2) Defining a semi-order relation on X, we have a semi-order structure T. Anyway, we can introduce a corresponding semi-order to any system which can be represented by a directed acyclic graph. Hence, semi-order relation represents a wide range of structures in real world.
(3) The structure of X can be induced by some operations defined on X. For example, given a unary operator 
, i.e., 
, it implies that a relation between elements x and 
is established. Linking these two elements by a directed edge, we define a directed graph corresponding to operator p. And, the directed graph determines a corresponding pseudo semi-order relation as well. If p is a multivariate operator on X, 
, it can also be represented by a graph. For example, each 
connects to y with a directed edge. And the n directed edges connected to y are linked by a curved arc. Then, we have a graph corresponding to p and it is somewhat similar to an AND/OR graph or a hyper-graph. From the graph, we can define its structure, T. So the relation among elements can be defined by multivariate operation p as well.
, i.e., , it implies that a relation between elements x and is established. Linking these two elements by a directed edge, we define a directed graph corresponding to operator p. And, the directed graph determines a corresponding pseudo semi-order relation as well. If p is a multivariate operator on X, , it can also be represented by a graph. For example, each connects to y with a directed edge. And the n directed edges connected to y are linked by a curved arc. Then, we have a graph corresponding to p and it is somewhat similar to an AND/OR graph or a hyper-graph. From the graph, we can define its structure, T. So the relation among elements can be defined by multivariate operation p as well.This section only tackles the problem of constructing [f] under topologic structures. For semi-order structures, see discussion in Chapter 3.
Assume that 
and 
are two topologic spaces. 
is an attribute function on X. The continuity of f is a key issue in analyzing the properties of 
. For example, we are given a premise 
, in order to affirm if the conclusion 
holds or not, our objective is to determine if there exists a sequence 
of reasoning steps from 
to 
. If so, the conclusion 
holds, otherwise, 
does not hold. This is similar to the concept of ‘path-connected’ in topology. We show the concept as follows. If 
, there exist 
such that each line segment 
belongs to A, then 
and 
are said to be path-connected in A. Therefore, by introducing a proper topology T into X, the problem that if 
can infer 
is transformed to that of whether 
and 
belong to the same path-connected component. In Section 2.5, we’ll discuss the problem in more details.
and are two topologic spaces. is an attribute function on X. The continuity of f is a key issue in analyzing the properties of . For example, we are given a premise , in order to affirm if the conclusion holds or not, our objective is to determine if there exists a sequence of reasoning steps from to . If so, the conclusion holds, otherwise, does not hold. This is similar to the concept of ‘path-connected’ in topology. We show the concept as follows. If , there exist such that each line segment belongs to A, then and are said to be path-connected in A. Therefore, by introducing a proper topology T into X, the problem that if can infer is transformed to that of whether and belong to the same path-connected component. In Section 2.5, we’ll discuss the problem in more details.Generally, X is rather complicated. It is not easy to judge whether any two given points belong to the same path-connected component. By using transformation function 
, the problem can be transferred into an easier one if the structure of Y is simpler than X. Obviously; we expect that if set 
is a connected, the transformed set 
is still connected. From topology, we know that if 
is a connected set, the image 
of A is also a connected set in Y when f is continuous. Once again it shows that the continuity of function f is a very significant property.
, the problem can be transferred into an easier one if the structure of Y is simpler than X. Obviously; we expect that if set is a connected, the transformed set is still connected. From topology, we know that if is a connected set, the image of A is also a connected set in Y when f is continuous. Once again it shows that the continuity of function f is a very significant property.By partitioning 
we have a quotient space 
. From f we induce a function 
on [X]. Obviously, we expect that [f] would still be continuous. Unfortunately, it is not true generally. Even so we expect that [f] still remains some weaker continuity than the strict one in favor of the analysis.
we have a quotient space . From f we induce a function on [X]. Obviously, we expect that [f] would still be continuous. Unfortunately, it is not true generally. Even so we expect that [f] still remains some weaker continuity than the strict one in favor of the analysis.1. Constructing 
First, we introduce some concepts below.
Definition 2.5
Assume that 
is a metric space. Let 
, 
.
is a metric space. Let , .
is said to be a diameter of A.Definition 2.6
R is an equivalence relation on 
. Let 
, where [X] is a quotient space corresponding to R. 
is said to be fineness of R. Intuitively, 
can be regarded as ‘the maximal diameter’ of equivalence classes corresponding to R.
. Let , where [X] is a quotient space corresponding to R. is said to be fineness of R. Intuitively, can be regarded as ‘the maximal diameter’ of equivalence classes corresponding to R.Definition 2.7
Assume that 
, R is an equivalence relation on 
, and [X] is a quotient space under R. Let 
. 
is said to be fineness of R with respect to f.
, R is an equivalence relation on , and [X] is a quotient space under R. Let . is said to be fineness of R with respect to f.Definition 2.8
Assume that 
. Let 
. M is called the ratio of expansion and contraction of f, or simply REC.
. Let . M is called the ratio of expansion and contraction of f, or simply REC.Definition 2.9
Assume that 
, where 
is a metric space. Given a positive constant 
, if 
there exists an open set u(x) such that 
. f is said to be 
-continuous on X, where 
.
, where is a metric space. Given a positive constant , if there exists an open set u(x) such that . f is said to be -continuous on X, where .Proposition 2.10
Assume that 
is a continuous function, [X] is a quotient space of X. Define 
as 
, where 
.
is a continuous function, [X] is a quotient space of X. Define as , where .Suppose that M is a REC of f and the fineness of [X] is l. Given 
, we obtain that 
is a 
-continuous function on X, where 
is a natural projection function.
, we obtain that is a -continuous function on X, where is a natural projection function.Proof:
, from the continuity of f, there exist 
and 
such that 
.Since the fineness of [X] is l, we can write the diameter of 
.
.From the REC M of f, we have the diameter of 
From the definition of [f], we obtain
That is
Given 
by letting 
be small enough such that 
, we have that 
is 
-continuous at x. 
is 
-continuous on X.
by letting be small enough such that , we have that is -continuous at x. is -continuous on X.
-Continuous FunctionExample 2.11
As shown in Fig. 2.1, 
is a continuous function.
is a continuous function.Assume that 
. Axis X is partitioned into several intervals equally, each interval 
with length
. If using a step function as follows
. Axis X is partitioned into several intervals equally, each interval with length. If using a step function as follows
, 
, from Proposition 2.10, we can see that 
is 2Ml-continuous on X.Proposition 2.10 indicates that under quotient space transformation the continuity of the transformed function does not always hold but α-continuity, a weak continuity, is preserved. It implies that if viewing from a coarse granular level, the properties of the original function become rough (fuzzy). Intuitively, it may be likened to copperplate etching technique. The coarser the etching grain-size of a copper plate the rougher its image.
Generally speaking, instead of some attribute f in the original space, a new attribute 
can be used for describing the corresponding one in its quotient space. In order to transform the original attribute function f into 
, it is needed to redefine the function. As we know, the original function f(x) is defined on ‘all neighborhoods of x’ or ‘a neighborhood existed in x’, we now need to redefine 
-f(x) on ‘a fixed 
-neighborhood of x’ or ‘some fixed neighborhood of x’. This idea has also been used in the mathematical morphology method of pattern recognition (Serra, 1982).
can be used for describing the corresponding one in its quotient space. In order to transform the original attribute function f into , it is needed to redefine the function. As we know, the original function f(x) is defined on ‘all neighborhoods of x’ or ‘a neighborhood existed in x’, we now need to redefine -f(x) on ‘a fixed -neighborhood of x’ or ‘some fixed neighborhood of x’. This idea has also been used in the mathematical morphology method of pattern recognition (Serra, 1982).We might as well try to analyze two basic operations in mathematical morphology from our granular computing point of view.
Definition 2.10
Assume that X is a domain. 
, 
, a corresponding subset 
is given on X. 
is called a structural element with respect to x. Define
, , a corresponding subset is given on X. is called a structural element with respect to x. Define
Dilation and erosion are two basic operations in mathematical morphology. Any morphological transformation can be obtained through union, intersection and deference of these two operations.
In topology, the minimal closed set containing A is called a closure of A denoted by 
and is defined as:
is an open set containing x.
and is defined as:
is an open set containing x.Accordingly, given a set A, the maximal open set being contained by A is called an inner-kernel of A and denoted by 
. Mathematically, the definition is as follows:
. Mathematically, the definition is as follows:
Either closure or inner-kernel operation can be used for defining a topologic structure of a space. Hence, closure and inner-kernel operations are two basic operations in topology.
It is clear that both 
and D(A) are quite similar. So long as replacing ‘
’ (in the definition) by a fixed subset B(x), the concept 
becomes D(A). So the concept D(A) is equivalent to ‘α-closure’. The same is true for the relation between 
and E(A). Thus, E(A) is equivalent to ‘α-inner kernel’.
and D(A) are quite similar. So long as replacing ‘ ’ (in the definition) by a fixed subset B(x), the concept becomes D(A). So the concept D(A) is equivalent to ‘α-closure’. The same is true for the relation between and E(A). Thus, E(A) is equivalent to ‘α-inner kernel’.We know that D(A) and E(A) are the rough descriptions of closure 
and inner-kernel 
in the coarse granular level, respectively. The aim of mathematical morphology is to extract some essential characteristics of images while ignoring their details. This is identical with the ‘α-∗∗’ concept. As J. Serra pointed out ‘The images under study exhibit too much information and the goal of any morphological treatment is to manage the loss of information through the successive transformations’ (Serra, 1982). The multi-granular computing is similar to the above idea.
and inner-kernel in the coarse granular level, respectively. The aim of mathematical morphology is to extract some essential characteristics of images while ignoring their details. This is identical with the ‘α-∗∗’ concept. As J. Serra pointed out ‘The images under study exhibit too much information and the goal of any morphological treatment is to manage the loss of information through the successive transformations’ (Serra, 1982). The multi-granular computing is similar to the above idea.The concept of ‘α-∗∗’ is an outcome of hierarchy. The similarity between the concept of ‘α-∗∗’ and the concept of dilation (or erosion) in mathematic morphology indicates that our quotient structure model can be extended to more general cases.
When space X is transformed to its quotient space [X], i.e., 
, where p is a natural projection, the attribute ‘∗∗’ becomes ‘α-∗∗’. Similarly, when 
, where 
is a subset of X, the 
(or 
) becomes D(A) (or E(A)). In this case, if X is an original space, 
can be regarded as a ‘quotient space’ of X by considering each 
as an element.
, where p is a natural projection, the attribute ‘∗∗’ becomes ‘α-∗∗’. Similarly, when , where is a subset of X, the (or ) becomes D(A) (or E(A)). In this case, if X is an original space, can be regarded as a ‘quotient space’ of X by considering each as an element.Definition 2.11
Assume that 
is a metric space. If X cannot be represented by the union of two non-empty sets A and B that satisfy the following condition (2.9)
is a metric space. If X cannot be represented by the union of two non-empty sets A and B that satisfy the following condition (2.9)
(2.9)
Property 2.1
Assume that 
, R is an equivalence relation on X, and 
. The ratio of expansion and contraction (or REC) of f is M. We have:
, R is an equivalence relation on X, and . The ratio of expansion and contraction (or REC) of f is M. We have:
Proof:
From the definitions of 
and 
, we have the proof directly.
and , we have the proof directly.Theorem 2.1
Assume that 
is continuous, X is connected, R is an equivalence relation on X, and 
. The REC of f is M. Define
is continuous, X is connected, R is an equivalence relation on X, and . The REC of f is M. Define
That is, we take on 
at any point 
as the value of 
at point 
. Thus, 
is 2Ml-connected.
at any point as the value of at point . Thus, is 2Ml-connected.Proof:
Let 
. From Property 2.1, we have 
.
. From Property 2.1, we have .Reduction to absurdity, assume that 
is not d-connected. Letting 
, there exist non-empty sets A and B that satisfy (Fig. 2.2):
is not d-connected. Letting , there exist non-empty sets A and B that satisfy (Fig. 2.2):
We first show that
Since
That is,
Similarly
Since 
and 
, by letting 
, again from 
and 
, we have 
.
and , by letting , again from and , we have .Similarly, we have 
.
.From 
. 
and 
are closed sets and f is continuous, we know that 
and 
are closed on X. Therefore, 
, where 
and 
are non-empty closed sets and 
. Thus, X is not connected, which contradicts the assumption that X is connected.
. and are closed sets and f is continuous, we know that and are closed on X. Therefore, , where and are non-empty closed sets and . Thus, X is not connected, which contradicts the assumption that X is connected.Consequently, 
is d-connected.
is d-connected.The concept of d-connected can be regarded as a description of the degree of connectivity of a set. The smaller the d, the closer to true connectivity the d-connectivity is. In other words, regarding any two sets on X as being connected provided their distance is less than d, then we have d-connected. The theorem indicates that if the REC of f is fixed, the roughness of connectivity of images in the coarse granular level is inversely proportional to the fineness of R. That is, the finer the partition, the less rough (finer) the connectivity of images in the coarse granular level. Conversely, keeping the fineness of R fixed, the roughness of connectivity is proportional to the REC of f. That is, the larger the REC the rougher the connectivity of the mapped images. The above intuition can accurately be proved by our quotient space model.
Obviously, the concept of d-connectivity is an instantiation of the concept of ‘
∗∗’ that we mentioned before. After the partition, some information must be lost in coarse granular levels. Generally, it is hard to preserve the original attributes (e.g., continuity, connectivity, etc.) of the original space. By introducing the concept of ‘
∗∗’ attributes, weaken attributes, the original attributes will remain valid in a relaxed sense. This will provide a useful clue to the analysis of coarse granular levels.
∗∗’ that we mentioned before. After the partition, some information must be lost in coarse granular levels. Generally, it is hard to preserve the original attributes (e.g., continuity, connectivity, etc.) of the original space. By introducing the concept of ‘∗∗’ attributes, weaken attributes, the original attributes will remain valid in a relaxed sense. This will provide a useful clue to the analysis of coarse granular levels.For some concept such as connectivity in topology, a set either is connected or not. Either of the two facts must be true. By introducing ‘d-connectivity’, the different degrees of connectivity can be described. This is just like the concept of membership functions in fuzzy mathematics. The concept of ‘
∗∗’ attributes that we presented here virtually relates our methodology to that of fuzzy mathematics. It makes granular computing more powerful.
∗∗’ attributes that we presented here virtually relates our methodology to that of fuzzy mathematics. It makes granular computing more powerful.2. Constructing 
In the preceding section, we know that the value of 
can also be represented in quotient space 
of Y. We next present its properties.
can also be represented in quotient space of Y. We next present its properties.Assume that 
and 
are topologic spaces, 
is a one-to-one corresponding function, R is an equivalence relation on X. [X] is a quotient space of X with respect to R.
and are topologic spaces, is a one-to-one corresponding function, R is an equivalence relation on X. [X] is a quotient space of X with respect to R.Definition 2.12
Define an equivalence relation 
on Y such that:
on Y such that:
That is, x and y are equivalent on Y if and only if the original images of x and y are equivalent on X. 
is said to be an equivalence relation on Y induced from R via f or an induced equivalence relation from R.
is said to be an equivalence relation on Y induced from R via f or an induced equivalence relation from R.Conversely, if 
is an equivalence relation on Y, define an equivalence relation on X such that
via f or an induced equivalence relation from 
, where f is not necessarily one-to-one correspondent.
is an equivalence relation on Y, define an equivalence relation on X such that
via f or an induced equivalence relation from , where f is not necessarily one-to-one correspondent.Lemma 2.1
Assume that 
and 
are topologic spaces, 
, R is an equivalence relation on X, 
is an equivalence relation on Y induced from R. Let 
and 
be quotient spaces corresponding to R and 
, respectively. Define
and 
are natural projections. Then, for 
, we have 
.
and are topologic spaces, , R is an equivalence relation on X, is an equivalence relation on Y induced from R. Let and be quotient spaces corresponding to R and , respectively. Define
and are natural projections. Then, for , we have .Proof:
Obviously, 
.
.Conversely,
is a set composed by elements of [X], where the element of [X] is a subset of X.
is a set composed by elements of [X], where the element of [X] is a subset of X.Theorem 2.2
Assume that 
and 
are topologic spaces, R is an equivalence relation on X, 
is an equivalence relation on Y induced from R, 
and 
are quotient spaces with respect to R and 
, respectively, 
is a continuous function. Let 
be 
, where 
and 
are projection functions. Thus, 
is continuous.
and are topologic spaces, R is an equivalence relation on X, is an equivalence relation on Y induced from R, and are quotient spaces with respect to R and , respectively, is a continuous function. Let be , where and are projection functions. Thus, is continuous.Proof:
For 
, assume that 
is an arbitrary neighborhood of 
. Let 
. Regarding 
as a set of Y, it is open. Since f is continuous, we know that u is an open set on X. From Lemma 2.1, we can see that u is a set on [X]. Thus, together with the definition of quotient topology, it implies that u is a neighborhood of a. From the definition 
, we have that 
is continuous at a. That is, 
is continuous.
, assume that is an arbitrary neighborhood of . Let . Regarding as a set of Y, it is open. Since f is continuous, we know that u is an open set on X. From Lemma 2.1, we can see that u is a set on [X]. Thus, together with the definition of quotient topology, it implies that u is a neighborhood of a. From the definition , we have that is continuous at a. That is, is continuous.Corollary 2.1
If 
is an equivalence relation on Y and R is an equivalence relation on X induced from 
. 
is continuous. Then 
, for 
is continuous.
is an equivalence relation on Y and R is an equivalence relation on X induced from . is continuous. Then , for is continuous.Theorem 2.2 presents another approach for constructing function [f]. It has a wide-ranging application, for example, qualitative reasoning in Al. We next analyze the Example 2.11 in Section 2.3.1.
When an object is thrown upwards, its state at moment x can be represented by 
, where 
and 
indicate its distance from the ground and velocity at moment x, respectively. Now only the qualitative properties of its state are paid attention to. The range 
of 
is partitioned into two classes {0} and 
. While the range 
of 
is divided into three classes {0}, 
, and 
. 
is regarded as a continuous function on 
. The preceding partition corresponds to an equivalence relation 
on Y. From Theorem 2.2, in order to construct [f], we need an equivalence relation on X induced from 
.
, where and indicate its distance from the ground and velocity at moment x, respectively. Now only the qualitative properties of its state are paid attention to. The range of is partitioned into two classes {0} and . While the range of is divided into three classes {0}, , and . is regarded as a continuous function on . The preceding partition corresponds to an equivalence relation on Y. From Theorem 2.2, in order to construct [f], we need an equivalence relation on X induced from .
is partitioned into {0} and 
, 
is its induced equivalence relation on X. Then, 
: 
.R is partitioned into 
, {0} and 
, 
is its induced equivalence relation on X. Then,
indicates the inverse transformation of the first component of f,
, {0} and , is its induced equivalence relation on X. Then,
indicates the inverse transformation of the first component of f,
indicates the inverse transformation of the second component of f.The combination equivalence relation of 
and 
is 
and
and is and
Let 
and 
be quotient spaces with respect to 
and 
, respectively. From Theorem 2.2, we know that 
, where 
is a projection function, and [f] is a continuous function of 
.
and be quotient spaces with respect to and , respectively. From Theorem 2.2, we know that , where is a projection function, and [f] is a continuous function of .If the first component {0}, 
of [Y] is named as ‘0’, ‘+’; the second component 
and 
of [Y] as ‘–’, ‘0’ and ‘+’, we have [f]((t1, t2)) = (+, −), [f]({t1}) = (+, 0), etc. These results are consistent with that shown in Example 2.11 of Section 2.3.1.
of [Y] is named as ‘0’, ‘+’; the second component and of [Y] as ‘–’, ‘0’ and ‘+’, we have [f]((t1, t2)) = (+, −), [f]({t1}) = (+, 0), etc. These results are consistent with that shown in Example 2.11 of Section 2.3.1.From our quotient space model, a strong property of [f] is discovered, that is, [f] is a 
continuous function, if it is constructed in the way that we already showed. In the light of the result, we can see that this is one of the possible ways for partitioning X (or Y).
continuous function, if it is constructed in the way that we already showed. In the light of the result, we can see that this is one of the possible ways for partitioning X (or Y).2.3.4. Conclusions
In this section, we have discussed how to establish a function [f] on [X] induced from f. When X is an unstructured domain, we presented four basic methods for constructing [f], that is, statistics, closure, quotient space and combination methods. If X is a structured domain, only topologic structures and continuous function f are involved. When [f] is a 
function, we introduced the concepts of 
continuity and 
connectivity and established the corresponding properties of [f]. When [f] is a 
function, we presented an approach for constructing function [f] which guarantees its continuity.
function, we introduced the concepts of continuity and connectivity and established the corresponding properties of [f]. When [f] is a function, we presented an approach for constructing function [f] which guarantees its continuity.So far we have established all three elements of problem space 
. Further discussion will be presented in Chapters 3 and 4.
. Further discussion will be presented in Chapters 3 and 4...................Content has been hidden....................
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