Reduction of Knowledge

Contents

Introduction

Reduct and Core of Knowledge

Relative Reduct and Relative Core of Knowledge

Reduction of Categories

Relative Reduct and Relative Core of Categories

1.Introduction

• The fundamental problem in the rough set approach is discovering dependencies between attributes in an Information System because it enables to reduce the set of attributes removing those that are not essential (unnecessary) to characterize knowledge.

• This problem called knowledge reduction.

2.Reduct and core of knowledge

• ReductReduct:: a reduct of knowledge is its essential part.

• CoreCore:: is in a certain sense its most important part.

Another Definition:A reductreduct is the minimal subset of attributes which

provides the same quality of classification as the set of all attributes.

If the information table have more than one reduct, the intersection of all of them is called the core

Dispensable & Indispensable

• Let R be a family of equivalence relations• let • if IND(R) = IND(R-{a}), then a is dispensable in

R• if IND(R) ≠ IND(R-{a}), then a is indispensable in

R

• the family R is independent if each is indispensable in R ; otherwise R is dependent

Ra

Ra

Dispensable means that the relation can be excluded andthe Indiscernibility of the relations R will not change Indispensable means that the relation can’t be excludedbecause the Indiscernibility of the relations R will change

The set of all indispensable relation in R => the core of R, CORECORE(R)(R) )

is a reduct of R if Q is independent and IND(Q) = IND(R) , ( REDRED(R)(R) )RQ

)R()R( REDCORE

)R()R( REDCORE

Example 1

R={a, c}

U/{a}={{ 1, 2, 6}{3, 4}{5, 7}}

U/{c}={{ 2}{3, 4}{5, 6, 7}{1}}

U/IND(R) ={{1}{ 2}{6} {3,4}{5,7}}=U/{a,b}

∴ a, c : indispensable in R

∴ R is independent

(As U/R ≠ U/{b}, U/R ≠ U/{a})

UU a c1 1 4

2 1 1

3 2 2

4 2 2

5 3 3

6 1 3

7 3 3

Proposition

• If R is independent and P R then P is also independent.

Example2

a family of equivalence relations R={P, Q, R}• U/P = {{1,4,5}{2,8}{3}{6,7}}• U/Q = {{1,3,5}{6}{2,4,7,8}}• U/R = {{1,5}{6}{2,7,8}{3,4}}

Thus the relation IND(R) has the equivalence classes• U/R={{1,5}{6}{2,8}{3}{4}{7}}

U/{P,Q} = {{1,5}{4}{{2,8}{3}{6}{7}}

U/{P,R} = {{1,5}{4}{2,8}{3}{6}{7}}

U/{Q,R} = {{1,5}{3}{6}{2,7,8}{4}}

U/{P,Q}= U/R =>R is dispensable in R

U/{P,Q}= U/R =>R is dispensable in R

U/{P,Q}= U/R =>R is dispensable in R

CORE CORE ( R ) = { ( R ) = { P P }}

RED RED ( ( R R ) = {) = {P,QP,Q} and {} and {P,RP,R}}

As U/{P,Q}≠U/{P} , U/{P,Q}≠U/{Q}U/{P,R}≠U/{P} , U/{P,R}≠U/{R}

3.Relative Core and Relative Reduct of Knowledge

• Here we will give a generalization of a concepts of reduct and core.• We define a concept of a positive region of a classification with

respect to another classification. • Let R and Q be families of equivalence relation over U,

if , then the attribute a is dispensable in P ,

if , then the attribute a is indispensable in P ,

The P-positive region of Q :

• If every c in R is D-indispensable, then we say that R is D-independent (or R is independent with respect to D)

XPOSUX

Q/

P P)Q(

)Q()Q( }){(P aPPOSPOS )Q()Q( }){(P aPPOSPOS

.Ra

The P-positive region of QP-positive region of Q is the set of all objects of U , which can be properly classified to classes of U/Q employing knowledge expressed by the classification U/P.

Take a closer look !

Example3

R={a, c} D={d}

U/{a} = {{ 1, 2, 6}{3, 4}{5, 7}}

U/{c} = {{ 2}{3, 4}{5, 6, 7}{1}}

U/D = {{1,4,6}{2,3,5,7}}

U/IND(R) ={{1}{ 2}{6} {3,4}{5,7}}=U/{a,c}

UU a c d1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

}7,6,5,2,1{}}7,5,2}{6,1{{R)D(D/

R

XPOSUX

}2,1{}}2}{1{{)D(}){R( aPOS

}7,5{}}7,5{{)D(}){R( cPOS

)D()D( }){R(R aPOSPOS

)D()D( }){R(R cPOSPOS

the relation ‘a’ is indispensable in R (‘a’ is indispensable attribute)

the relation ‘c’ is indispensable in R (‘c’ is indispensable attribute)

R is D-independent

)D()D( }){R(R aPOSPOS )D()D( }){R(R cPOSPOS

Now we are able to give definitions of generalized concepts considered previously:

• Let P and Q be families of equivalence relations over U , we say that R P is Q- dispensableR P is Q- dispensable in P , if

• Otherwise, R is Q-indispensableR is Q-indispensable in P.• If every R P is Q-indispensable, we will say that P is Q-P is Q-

independentindependent (or P is independent with respect to Q)

• The family S P will be called a Q-reduct of PQ-reduct of P , if and only if S is the Q-independent subfamily of P

• The set of all Q-indispensable elementary relations in P will be called the Q-core of PQ-core of P, and will be denoted as

)Q)(()Q)(( (R))-IND(PIND(P) INDPOSINDPOS

)Q()Q( PS POSPOS

)P(QCORE

Proposition

Where is the family of all Q-reducts of P.

Example:• U={U1, U2, U3, U4, U5, U6} =let {1,2,3,4,5,6}• Ω={headache, Muscle pan, Temp, Flu}={a, b, c, d}• condition R={a, b, c}, decision D={d}

)()( PREDPCORE QQ

)P(QRED

U/{a}={{1,2,3}{4,5,6}}U/{b}={{1,2,3,4,6}{5}}U/{c}={{1,4}{2,5}{3,6}}U/{a,b}={1,2,3}{4,6}{5}}U/{a,c}={{1}{2}{3}{4}{5}{6}}U/{b,c}={{1,4}{2}{3,6}{5}}U/R={{1}{4}{2}{5}{3}{6}}U/D={{1,4,5}{2,3,6}}

• POSR(D)={{1,4,5}{2,3,6}}={1,2,3,4,5,6}• POSR-{a}(D)={{1,4,5}{2,3,6}}= {1,2,3,4,5,6}• POSR- {b}(D)={{{1,4,5}{2,3,6}}= {1,2,3,4,5,6}• POSR-{c}(D)={{5}}={5} relation ‘a’, ‘b’ is dispensable relation ‘c’ is indispensable => D-core of R =CORED(R)={c}

U Headache Muscle pain

Temp. Flu

U1 Yes Yes Normal No U2 Yes Yes High Yes U3 Yes Yes Very-high Yes U4 No Yes Normal No U5 No No High No U6 No Yes Very-high Yes

To find reducts of R={a, b, c}

{a, c} is D-independent and POS{a,c}(D)=POSR(D)

( POS{a}(D)={} ≠POS{a, c}(D)

POS{c}(D)={1,4,3,6} ≠ POS{a, c}(D) ) {b,c} is D-independent and POS{b, c}(D)=POSR(D) => {a, c} {b, c} is the D-reduct of R

• POSR-{ab}(D)={{1,4}{3,6}}={1,4,3,6}• POSR-{ac}(D)={{5}}={5}• POSR-{bc}(D)={}

Some definitions:

Set POSP (Q) is the set of all objects which can be classified to elementary categories of knowledge Q , employing knowledge P.

Knowledge P is Q-independent if the whole knowledge P is necessary to classify objects to elementary categories of knowledge Q

The Q-CORE knowledge of P is the essential part of knowledge P , which can not be eliminated without disturbing the ability to classify objects to elementary categories of Q

The Q-reduct of knowledge P is the minimal subset of knowledge P , which provides the same classification of objects to elementary categories of knowledge Q as the whole knowledge of P

Knowledge P with only one Q-reduct is in a sense , deterministic

4.Reduction of categories

• Every basic category is "built up" (is an intersection) of some elementary categories.

• Thus the question arises whether all elementary categories are necessary to define the basic categories in question.

• This problem is similar to that of reducing knowledge ,i.e. elimination of equivalence relations which are superfluous to define all basic categories in knowledge P.

Definitions

Let F = { X1, …. , Xn } , be a family of sets such that Xi U.

• Xi is dispensable in F, if (F – {Xi}) = F ; otherwise

the set Xi is indispensable in F .

• The family F is independent if all of its components are indispensable in F; otherwise F is dependent.

• The family H F is a reduct of F, if H is independent and H = F

• The family of all indispensable sets in F will be called the CORE of F , denoted CORE(F)

Proposition

CORE (F) = RED (F)Where RED (F) is the family of all reducts of F.

Example: – Suppose we are given the family of three sets F ={X, Y, Z}, where

X = {1, 3, 8} Y = {1, 3, 4, 5, 6} Z = {1, 3, 4, 6, 7} Hence F = X Y Z = {1, 3},Because (F – {X}) = Y Z = {1, 3, 4, 6} (F – {Y}) = X Z = {1, 3} (F – {Z}) = X Y = {1, 3}

Sets Y and Z are dispensable in the family F,hence the family F is dependent,

{X,Y} and {X,Z} are {X,Y} and {X,Z} are reductsreducts of F, of F, and and

{X,Y} {X,Z} = {X} is the {X,Y} {X,Z} = {X} is the corecore of F. of F.

5.Relative Reduct and Core of Categories

We will generalize the concept of the reduct, and the core, with respect to a specific set.

Suppose we are given a family { X1, …. , Xn } , Xi U and a subset Y U such that F Y.

We say that Xi is Y-dispensable in F , if (F – {Xi}) Y ; otherwise the set Xi is Y-indispensable in F

The family F is Y-independent in F if all of its components are Y-indispensable in F; otherwise F is Y-dependent.The family H F is a Y-reduct of F, if H is Y-independent in F and H Y

The family of all Y-indispensable sets in F will be called the Y- core of F and will be denoted COREY (F) .

We will also say that a Y-reduct (Y-Core) is a relative reduct (Core)relative reduct (Core) with respect to Y.

Proposition

COREY (F) = REDY (F)Where REDY (F) is the family of all Y- reducts of F.

Example:Suppose we are given the family of three sets F ={X, Y, Z},where X = {1, 3, 8} Y = {1, 3, 4, 5, 6} Z = {1, 3, 4, 6, 7} and F = X Y Z = {1, 3} Let T = {1, 3, 8} F .

we are able to see whether sets X, Y, Z are T-dispensable in the family F or not.

(F – {X}) = Y Z = {1, 3, 4, 6}(F – {Y}) = X Z = {1, 3}(F – {Z}) = X Y = {1, 3}

Hence set X is T-indispensable, sets Y and Z are T-dispensable,Family F is T-dependentThe T-core of F is set X, and there are two T-reducts of the family F, {X, Y} and {X, Z}.