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Exploring Reducts in Incomplete Information Systems: Definitions and Examples

This document discusses the concept of reducts in incomplete information systems, specifically focusing on the definitions, examples, and computations of reducts from decision systems. It explores similarity relations, tolerance classes, and illustrates how to derive reducts for classification attributes with examples such as price, size, and accident history. The text emphasizes the significance of both relative reducts and decision systems in the realm of information systems. Key computations and discernibility functions are provided to enhance understanding.

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Exploring Reducts in Incomplete Information Systems: Definitions and Examples

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  1. REDUCTS IN INCOMPLETE INFORMATION SYSTEMS ZbigniewRas

  2. Information Systems S = (X, AT) isan information system, where • X - objects, • AT-attributes (partial functions from X into 2Va {*}), • Va - set of values of attribute a.

  3. 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)  XX: (a A)[a(x)  a(y)  or a(x) = * or a(y) = *]}. •  SIM(A) is a tolerance relation (reflexive, symmetric). •  Let IA(x) = {yX: (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.

  4. Definition: A  AT is a reductof information system S = (X, AT) iff • SIM(A) = SIM(AT) and • (BA)[SIM(B)  SIM(A)].  A  AT is a reduct of information system S=(X, AT) for x iff • IA(x) = IAT(x) and • (BA)[IB(x)  IA(x)].  In our example {Price, Size, Accident} is a reduct of S.

  5. Decision Systems • 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/

  6. Example 2:Decision System S with “generalized decision” as the extra feature. Definition: Set A  AT is a reduct of S (relative reduct or d-reduct), iff A = AT and (BA)[ B A ]. Set A  AT is a reduct of S for x  X (relative reduct for x or d-reduct for x) iff A(x) = AT(x) and (BA)[ B(x)A(x) ].

  7. 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)

  8. Computation of Reducts: Discernibility table Here we use P – Price, M – Mileage, S – Size, A- Accident. Discernibility Function: F(P,M,S,A) = PSA(PA) = PSA – (reduct) Reducts for objects: F(1)=PS, F(2)=PA, F(3)=SA, F(4)=PS, F(5)=SA, F(6)=PS.

  9. Discernibility table Computing d-Reducts: Here we use P – Price, M – Mileage, S – Size, A- Accident. Discernibility Function: F(P,M,S,A) = PSA(PA) = PSA – (d-reduct) d-reducts for objects: F(1)=PS, F(2)=PA, F(3)=SA, F(4)=(PA)S, F(5)=SA, F(6)=S.

  10. Thank You !

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