Chap3 Reduction of Knowledge

- Dongyi Jia- CS267 ID:104- Fall 2008

Agenda

Reduct and Core of KnowledgeRelative Reduct and Relative Core

of KnowledgeReduction of CategoriesRelative Reduct and Core Categorie

sSummary

Reduct and Core of Knowledge(1)

Two conceptsReduct: a reduct of knowledge is its essential

part, which suffices to define all basic concepts occuring in the considered knowledge.

Core: the core is in a certain sense its most important part

Reduct and Core of Knowledge(2)

Let R be a family of equivalence relations and Let R R. We will say that R is dispensable in R if IND(R) = IND(R-{R}); otherwise, R is indispensable in R. The family R is independent if each R R is indispensable in R; otherwise R is dependent.

Proposition 3.1

3.1 If R is independent and P R, then P is also independent.

The proof is by contradiction.

● Q P is a reduct of P if Q is independent and IND(Q) = IND(P).

● The set of all indispensable relations in P will be called the core of P, and will be denoted CORE(P).

Proposition 3.2

3.2 CORE(P)= REDUCT(P) where REDUCT(P) is the family of all reducts

of P.

The use of the concept of the core is twofold. 1. It can be used as a basis for computation of all redu

cts, for the core is included in every reduct, and its computation is straightforward.

2. The core can be interpreted as the set of most characteristic part of knowledge, which can not be eliminated when reducing the knowledge.

Example1 (1)

Suppose we are given a family R = {P, Q, R} of three equivalence relations P, Q and R with the following equivalence classes:

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

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

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

Thus the relation IND(R) has the equivalence classes

U/IND(R) ={{x1,x5},{x2,x8},{x3},{x4},{x6},{x7}}

Example1 (2)

The relation P is indispensable in R, sinceU/IND(R- {P}) ={{x1,x5},{x2,x7,x8},{x3},{x4},{x6}}

U/IND(R)

For relation Q we haveU/IND(R- {Q})

={{x1,x5},{x2,x8},{x3},{x4},{x6},{x7}}=U/IND(R), thus the relation Q is dispensable in R.

Similarly for relation R U/IND(R- {R}) ={{x1,x5},{x2,x8},{x3},{x4},{x6},{x7}}=

U/IND(R), since the relation R is also dispensable in R.

Example1 (3): conclusion

In order to find reducts of the family R={P,Q,R} we have to check whether pairs of relations P, Q and P,R are independent or not.

Because U/IND({P,Q}) U/IND(Q) and U/IND({P,Q}) U/IND(P), hence the relations P and Q are independent, and consequently {P,Q} is a reduct of R.

In the same way we find that {P,R} is also a reduct of R. Thus there are two reducts of the family R, namely {P,Q} and {P,R} and {P,Q} {P,R}={P} is the core of R.

Relative Reduct and Relative Core of Knowledge (1)

♦ we need first to define a concept of a positive region of a classification with respect to another classification.

Let P and Q be families of equivalence relations over U. By P-positive region of Q, denoted we understand the set

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

XPUQPOSp QUx /)(

)(QPOS p

Relative Reduct and Relative Core of Knowledge(2)—generalized concepts

Let P and Q be families of equivalence relations over U. We say that R P is Q-dispensable in P,

if Otherwise,R is Q-indispensable in P.

If every R in P is Q-indispensable, we will say that P is Q-independent.

The family S P will be called a Q-reduct of P, if and only if Sis the Q-independent subfamily of P and POS (Q)=POS (Q).pS

))(())(( }){()( QINDPOSQINDPOS RPINDPIND

Relative Reduct and Relative Core of Knowledge(3)

The set of all Q-indispensable elementary relations in P will be called the Q-core of P, and will be denoted as CORE (P).

It is easily seen that if P = Q we get the definitions introduced in the previous section.

q

Example2 (1)

Consider family R = {P, Q, R} of three equivalence relations P, Q and R with the following equivalence classes:

U/P ={{x1,x3,x4,x5,x6,x7},{x2,x8}}U/Q={{x1,x3,x4,x5},{x2,x6,x7,x8}}U/R ={{x1,x5,x6},{x2,x7,x8},{x3,x4}}.

The family R includes classificationU/IND(R) ={{x1,x5},{x2,x8},{x3,x4},{x6},{x7}}

Example2 (2)

Assume that the equivalence relation S is given with the equivalence classes

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

The positive region of S with respect to R is the union of all equivalence classes of U/IND(R) which are included in some equivalence classes of U/S, i.e. the set

POS (S) ={x1,x3,x4,x5,x6,x7}}.R

Proposition 3.3

3.3 CORE (P) = RED (P)

where RED (P) is the family of all Q-reducts of P.

Q QQ

Example2(continue)

In order to compute the core and reducts of with respect to S, we have first to find out whether the family R is S-dependent or not. According to definitions given in this section, we have to compute first whether P, Q and R are dispensable or not with respect to S (S-dispensable). Removing P we get

U/IND(R- {P}) ={{x1,x5},{x3,x4},{x2,x7,x8},{x6}}

Because

The P is S-indispensable in R,

Dropping Q from R we get

U/IND(R- {Q}) ={{x1,x5,x6},{x3,x4},{x2,x8},{x7}}

which yields the positive region

Hence Q is S-dispensable in R

)(}6,5,4,3,1{)(/ }){( SPOSxxxxxSINDU RPR

)(}7,6,5,4,3,1{)(}){( SPOSxxxxxxSPOS RQR

Example2(Continue)

Finally omitting R in R we obtain

U/IND(R- {R}) ={{x1,x3,x4,x5},{x2,x8},{x6,x7}},

and the positive region is

which means R is S-indispensable in R.

Thus the S-core of R is the set {P,R}, which is also the S-reduct of R.

)()(}){( SPOSSPOS RRR

Brief Comment

♦ Set POS (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 Q as a whole knowledge P.

p

Reduction of Categories(1)

From the mathematical point of view, 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.

The problem can be formulated precisely as follows.

Let F=X1,...,Xn, be a family of sets such that

We say that Xi is dispensable in F, if otherwise the 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 is a reduct of F, if H is independent and

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

UX i

FxF i }){(

FH FH

Reduction of Categories(2)

Prosposition 3.4

CORE(F) = RED(F)

where RED(F) is the family of all reducts of F.

Example3

Suppose we are given the family of three sets F = {X, Y, Z}, whereX = {x1,x3,x8}Y = {x1,x3,x4,x5,x6}Z = {x1,x3,x4,x6,x7}

Hence F = X Y Z = {x1, x3}Because (F -{X}) = Y Z = {x1, x3, x4, x6} (F -{Y}) = X Z = {x1, x3} (F -{Z}) = X Y = {x1, x3}sets Y and Z are dispensable in the family F, hence the family F isdependent, set X is the core of F, families {X,Y} and {X,Z} arereducts of F, and {X,Y} {X,Z\}={X} is the core of F.

Relative Reduct and Core Categories(1)

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

we say that is Y-dispensable in , if ; otherwise the set Xi is Y-indispensable in .

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

The family is a Y-reduct of , if H is Y-independent in and .

The family of all Y-indispensable sets in will be calledthe Y-core of F, and will be denoted . We will also saythat a Y-reduct (Y-core) is a relative reduct (core) with respect to Y.

UX i UY YF

F YXF i }){(F

FF

FH F

F

YH F

)(FCOREY

F

Relative Reduct and Core Categories(2)

Prosposition 3.5 CORE (F) = RED (F)

where RED (F) is the family of all Y-reducts of F.

Thus superfluous elementary categories can be eliminated from the basic categories in a similar way as the equivalence relations discussed in the previous section, in connection with reduction of relations.

Let us also note that if we obtain the case considered at the beginning of this section.

Y YY

FY

Example4

Consider another example of the family of three sets F = {X,Y,Z} whereX ={x1, x3, x8}Y ={x1, x3, x4,x5, x6}Z ={x1, x3, x4,x6, x7}And F = X Y Z = {x1,x3}

Let T = {x1, x3, x8} F. Now we are able to see whether set X, Y, Z are T-dispensable in the family F or not. we have to compute the following:

( F - {X}) = Y Z = {x1,x3,x4,x6}} ( F - {Y}) = X Z = {x1,x3} ( F - {Z}) = X Z = {x1,x3}

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

Summary

♦ Reducing of knowledge consists in removing of superfluous partitions (equivalence relations) or / and superfluous basic categories in the knowledge base.

♦ This procedure enables us to eliminate all

unnecessary knowledge from the knowledge base, preserving only that part of the knowledge which is really useful.

Thank you!