Computing Card-minimal Repairs - Repairing and Querying Databases under Aggregate Constraints - page 38

ρ 1 is the following:

a
1

ω : BalanceSheets ( 2008 , Receipts , cash sales , det , 1150 )= ⇒

χ 2 ( 2008 , 'cash sales' ) ≥ 1000

b

ω

1 : BalanceSheets

( , Receipts , cash sales , det , )= ⇒

2009 1110

χ ( 2009 , 'cash sales' ) ≥ 1000

2

a
2

ω : BalanceSheets ( 2008 , Receipts , receivables , det , 100 )= ⇒

χ 2 ( 2008 , receivables' ) ≤ 200

b

ω

2 : BalanceSheets

( , Receipts , receivables , det , )= ⇒

2009 90

χ ( , ) ≤

2009 receivables' 200

2

The set gr

( W ,ρ ( D ))

of ground weak aggregate constraints on the database re- 5

paired by ρ

5 is as follows:

c

ω

1 : BalanceSheets

( 2008 , Receipts , cash sales , det , 900 )= ⇒

χ 2 ( 2008 , 'cash sales' ) ≥ 1000

d
1

ω : BalanceSheets ( 2009 , Receipts , cash sales , det , 1110 )= ⇒

χ 2 ( 2009 , 'cash sales' ) ≥ 1000

c

ω

2 : BalanceSheets

( 2008 , Receipts , receivables , det , 350 )= ⇒

χ 2 ( 2008 , receivables' ) ≤ 200

d
2

ω : BalanceSheets ( 2009 , Receipts , receivables , det , 90 )= ⇒

χ 2 ( 2009 , receivables' ) ≤ 200

It is easy to see that all of the ground instantiations of ω 2 are satisfied by both ρ 1 ( D )

and ρ 5 ( D ) . Moreover, ρ 1 ( D ) also satisfies both of the instantiations of ω 1 , whereas

d

ρ 5 ( D ) satisfies only one instantiation of ω 1 (namely, ω

1 ). Hence,

γ ( ρ 1 ,W , D )= 0,

since ρ 1 ( D ) satisfies all of the ground aggregate constraints derived from ω 1 and

c

ω 2 , and γ ( ρ 5 ,W , D )= 1, since ρ 5 ( D ) violates the ground aggregate constraints ω

1 .

Thus, ρ 1 is a preferred repair w.r.t. AC and W .

4.5 Computing Preferred Repairs

Before tackling the problem of computing preferred card -minimal repairs, we in-
vestigate some issues related to the existence of preferred repairs from a theoretical
standpoint. The following theorem extends some of the results presented in Chap-
ther 3 to the case of preferred repairs.