Structure-Based Attribute Reduction: A Rough Set Approach - Feature Selection for Data and Pattern Recognition - page 131

Fig. 7.2 Strong-weak
hierarchy of 9 types of
structure-based reducts in
VPRSM

strong

weak

Proposition 1 Consider two types of reducts, ♥ -reducts and ♠ -reducts, and the

composition of them: ♥♠ -reducts. Let H and S be the set of all ♥ -reducts and

the set of all ♠ -reducts, respectively. Then the set of all ♥♠ -reducts is the set of all

minimal elements of { A ∪ B | A ∈ H and B ∈ S } .

7.3.4 Boolean Functions Representing Reducts

As shown above, L- and U-reducts in the classical RSM are characterized by prime
implicants of certain Boolean functions. In this section, we discuss Boolean functions
of 9 types of reducts in VPRSM. To do this, we focus on Boolean functions of reducts
pertaining to the lower approximations, the upper approximations, the boundaries,
the positive region, and the unpredictable region, since the others can be obtained by
taking conjunctions of those Boolean functions or using Proposition 1 .
First, we represent the preserving conditions by the generalized decision functions.

Lemma 3 be an admissible error rate, and A be a subset of C . We

have the following statements:

Let ʲ ∈[ 0 , 0 . 5 )

• Condition ( VPL ) with ʲ is equivalent to:

ʻ A ( ) = ʻ C ( u u ) for all u ∈ U . (VPLG)

• Condition ( VPU ) with ʲ is equivalent to:

˅ A ( ) = ˅ C (

u u ) for all u ∈ U . (VPUG)

• Condition ( VPB1 ) with ʲ is equivalent to:

(˅ \ ʻ) A ( ) = (˅ \ ʻ) C (

u u ) for all u ∈ U . (VPBG1)

• Condition ( VPP1 ) with ʲ is equivalent to:

ʻ A ( ) =∅⃔ ʻ C (

u u ) =∅ for all u ∈ U . (VPPG1)