REDUCTS IN

INCOMPLETE INFORMATION SYSTEMS

Zbigniew Ras

S = (X, AT) is an information system, where

X - objects, AT-attributes (partial functions from X into 2Va {*}), Va - set of values of attribute a.

Information Systems 

Example 1:

S = ({1,2,3,4,5,6}, {Price, Mileage, Size, Accident}) defined below:

Let A AT. By similarity relation based on A we mean:

 SIM(A) = {(x,y) XX: (a A)[a(x) a(y) or a(x) = * or a(y) = *]}.

 SIM(A) is a tolerance relation (reflexive, symmetric).

 Let IA(x) = {yX: (x,y) SIM(A)} - tolerance class for x with regard to A.

 X/SIM(A) = {IA(x) : x X} – not a partition of X in general.

 

Car Price Mileage Size Accident

1 {high} {high} {full} {doors, engine}

2 {low} * {full, compact} {engine}

3 * * {compact} {doors}

4 {high} * {full} {doors}

5 * * {full} {doors}

6 {low} {high} {full} *

A AT is a reduct of information system S = (X, AT) iff SIM(A) = SIM(AT) and (BA)[SIM(B) SIM(A)].

 A AT is a reduct of information system S=(X, AT) for x iff IA(x) = IAT(x) and (BA)[IB(x) IA(x)].

 In our example {Price, Size, Accident} is a reduct of S. 

Definition:

S = (X, AT {d}) decision system, where X - objects, AT - classification attributes, d - decision attribute, where d(x) Vd (value is certain).

Let A AT and A(x) = {v : d(y) = v and y IA(x)}

/generalized decision in S/

Decision Systems

Example 2:Decision System S with “generalized decision” as the extra feature.

Car Price Mileage Size Accident d AT

1 {high} {high} {full} {doors, engine} good {good, excel}

2 {low} * {full, compact} {engine} good {good}

3 * * {compact} {doors} poor {poor}

4 {high} * {full} {doors} good {good, excel}

5 * * {full} {doors} excel {good, excel}

6 {low} {high} {full} * good {good, excel}

Definition:Set A AT is a reduct of S (relative reduct or d-reduct), iff A = AT and (BA)[ B A ]. Set A AT is a reduct of S for x X (relative reduct for x or d-reduct for x) iffA(x) = AT(x) and (BA)[ B(x) A(x) ]. 

Example: {Size, Accident} is a is a relative reduct of S . {Size, Accident} is a relative reduct of S for object 3 and {Price, Accident} is a relative reduct of S for object 2. Relative reducts for objects are used to construct rules:

(Price, low) (Accident, engine) (d, good) (Size, compact) (Accident, doors) (d, poor) (Size, full) (d, good) (d, excel)

x/y 1 2 3 4 5 6

1

2 P

3 S A

4 P, A S

5 A S

6 P S P

Computation of Reducts:

Discernibility table

Here we use P – Price, M – Mileage, S – Size, A- Accident. Discernibility Function: F(P,M,S,A) = PSA(PA) = PSA – (reduct)

Reducts for objects: F(1)=PS, F(2)=PA, F(3)=SA, F(4)=PS, F(5)=SA, F(6)=PS.

Computing d-Reducts:

x/y 1 2 3 4 5 6

1

2 P

3 S A

4 PA S

5 A S

6 S

Discernibility table

Here we use P – Price, M – Mileage, S – Size, A- Accident.

Discernibility Function: F(P,M,S,A) = PSA(PA) = PSA – (d-reduct)

d-reducts for objects: F(1)=PS, F(2)=PA, F(3)=SA, F(4)=(PA)S, F(5)=SA, F(6)=S. 

Thank You !