Knowledge

Contents

Introduction

Knowledge and Classification

Knowledge Base

Equivalence of Knowledge

Generalization and Specialization of Knowledge

1.Introduction

The rough set Theory was firstly developed by Pawlak in the 1980s as a mathematical tool to deal with the uncertainty or vagueness inherent in a decision making process.

It is a useful notion for the classification of objects when the available information is not adequate to represent classes using precise sets.

Knowledge is a body of Information about some parts of reality which constitute domain of interest.

Figure 1

The knowledge here is close to that considered on Area of Cognitive Science discussed on AI.

Our theoretical framework here is rough sets for handling the Imprecise Knowledge or to deal with the Vagueness of knowledge.

2.Knowledge and Classification

KnowledgeKnowledge about the environment is primarily manifested to classify a variety of objects from the point of view of the survival in the real world.

Example1: • Complex classification patterns of sensor signals

Classification is the key issue in reasoning and decision making.

Knowledge consists of a family of various classification patterns of a domain of interest, which provide explicit facts about reality - together with reasoning capacity able to deliver implicit facts deliverable by explicit knowledge.

Example 2:– As we see in table 1 ,the knowledge enables us to

give the decision about each object in the universe.

Table 1: Walk: An example decision table

26-49 46-60 x7

26-49 16-30 x6

26-49 46-60 x5

1-25 31-45 x4

1-25 31-45 x3

016-30 x2

5016-30 x1

LEMSAge

no

yes

no

yes

no

no

yes

Walk

3.Knowledge Base

Suppose U is a finite set U ≠ Φ called Universe.

Any subset X U of the universe will be called a concept or a category in U.

Knowledge is a family of in concepts U.

Knowledge Base is a family of classifications over U.

Classification consists of concepts or categories of U.

Classification is called partition.

Figure 2: Universe of Discourse

U

Classification 1

Classification 2

Figure 3

C1

C5

C7 C8 C9

C2 C3

C4 C6

C1

C4C3

C2C1 C2 =

Some definitions: R: an equivalence relation over U U/R: the family of all equivalence classes

of R (or classification of U) referred to as categories or concepts of R

[x]R: a category in R containing an element x U

Equivalence Relation is Equivalence Relation is

classification,classification,

So we will use RSo we will use R

More definitions:

As we said before knowledge base is a set of classifications (Equivalence Relations).

if we have relation P and relation Q classify U to two equivalence families of categories and R is a set of all equivalence relations that classify U such that R ={P,Q} over U.

K=(U,R) is a relational system where U ≠ .

Indiscernibility relationIndiscernibility relation

If P R and P ≠ , then P (intersection of all equivalence relations belonging to P) is also an equivalence relation, which is denoted by IND(P), called an indiscernibility relation over P.

[x]IND (P) = [x]R .

RP U/IND(P) :the family of all equivalence classes of

the equivalence relation IND(P)

For more Clarification Example 3:

X2X1

X6

X5

X3

X4

X1 X2

X4 X5

X6

X3

X1 X2

X3

X5X6

X4

U/R1={{X1},{X2,X3,X4},{X5,X6}}

U/R2={{X1,X3,X4,X6},{X2,X5}}

U/R3={{X1,X3,X4,X5,X6},{X2}}

P is a set of relations over U, P ={R1,R2,R3} P=R1 R2 R3 = IND(P) and gives U/IND(P)

X1 X2

X4 X5

X6

X3U/IND(P)={{X1},{X2},{X3,X4},{X5},{X6}}[x]IND (P) = [x]R .

RP

Clarifying Example 4

If we look at the previous example we will see that

[x2]R1={X2,X3,X4}

[x2]R2={X2,X5}

[x2]IND(P) ={X2}

[x2]R1[x2]R2={{X2,X3,X4} {X2,X5}} = {X2}

[x2]IND(P)= [x2]RPR

Definitions:

• P-basic knowledge about U in K: denotes knowledge associated with the family of equivalence relations P.

• To simplify: use U/P instead of U/IND(P) and P is also called P-basic knowledge

• Basic categories (concepts) of knowledge P: equivalence classes of IND(P).

• Q-elementary knowledge: if Q R, then Q is called Q-elementary knowledge (about U in K)

• Q-elementary concepts (categories) of knowledge R: equivalence classes of Q

Example:

(Old), (ill) are Elementary Categories in K.

(Old ill) are Basic Categories in K.

• The family of basic categories in knowledge base K = (U, R): the family of all P-basic categories for all ≠ P R

• IND(K): the family of all equivalence relations defined in knowledge base K = (U, R)IND(K) = {IND(P) : P R}

• IND(K) is the minimal set of equivalence relations, containing all elementary relations of K and closed under set theoretical intersection of equivalence relations.

• K-categories: the family of all categories in the knowledge base K = (U, R)

Example5

Set of toy blocks U = {x1, x2, x3, x4, x5, x6, x7, x8} can be classified by color, shape and size

• x1, x3, x7 – are red,• x2, x4 – are blue,• x5, x6, x8 – are yellow,

• x1, x5 – are round,• x2, x6 – are square,• x3, x4, x7, x8 – are triangular,

• x2, x7, x8 – are large,• x1, x3, x4, x5, x6 – are small.

By these classifications we defined three equivalence relations R1, R2 and R3 having the following equivalence classes

U/R1 = { {x1, x3, x7}, {x2, x4}, {x5, x6, x8} }

U/R2 = { {x1, x5}, {x2, x6}, { x3, x4, x7, x8} }

U/R3 = { {x2, x7, x8}, {x1, x3, x4, x5, x6} }

Which are elementary concepts (categories) in our knowledge base K = (U, { R1 , R2 , R3 } ).

Basic categories are set theoretical intersections of elementary categories. For examples sets:{x1, x3, x7} {x3, x4, x7, x8} = {x3, x7}{x2, x4} {x2, x6} = {x2}{x5, x6, x8} {x3, x4, x7, x8} = {x8}are { R1, R2 }-basic categories red and triangular, blue and square, yellow and triangular respectively.Sets: {x1, x3, x7} {x2, x4} = {x1, x2, x3, x4, x7}{x2, x4} {x5, x6, x8} = {x2, x4, x5, x6, x8}{x1, x3, x7} {x5, x6, x8} = {x1, x3, x5, x6, x7, x8}are R1- categories red or blue (not yellow), blue or yellow (not red), red or yellow (not blue) respectively.

4.Equivalence of Knowledge

Let K = (U, P) and K’ = (U, Q) be two knowledge bases.

K and K’ (P and Q) are equivalentequivalent, if both K and K’ have the same set of basic categories, and consequently – the set of all categories. This means that knowledge in knowledge bases K and K’ enables us to express exactly the same facts about the universe.

That means if U/P = U/Q, K and K’ are equivalent

K ≃ K’ ,if IND (P) = IND(Q)

4.Generalization and Specialization of Knowledge

Let K = (U, P) and K’ = (U, Q) be two knowledge bases.

If IND(P) IND(Q) we say that: knowledge P (knowledge base K) is finer than

knowledge Q (knowledge base K’), or Q is coarser than P.

Generalization is combining some categories together Specialization lies in splitting categories into smaller

units.

if P is finer than Q,then P is specialization of Q,

and Q is generalization of P.

Example 6:

X1 X2

X4 X5

X6

X3

X1 X2

X4 X5

X6

X3

Q P Q is finer than PP is coarser than Q

U/Q U/P