Computing Card-minimal Repairs - Repairing and Querying Databases under Aggregate Constraints - page 36

the variables δ i assigned a value different from 0 are δ 3 and δ 18 , and variable w i

are assigned 0 except for w 3 = 250 and w 18 = 60. The card -minimal repairs cor-

ρ

responding to these solutions are = { t 2 , Value , 1150 , t 18 , Value , 1190 } and 1

ρ = { t 3 , Value , , t 18 , Value , }

350 1190 of Example 2.12. 5

4.4 Preferred Repairs

In general, there may be several card -minimal repairs for a database violating a
given set of aggregate constraints. For instance, in the “Balance Sheet” example,
it is possible to repair the data by increasing the values of attribute Value in tuples
t 2 and t 18 up to 1150 and 1190, respectively (this strategy is that adopted by the
card -minimal repair ρ 1 ). An alternative card -minimal repair consists of increasing

the values of attribute Value in tuples t 3 and t 18 up to 350 and 1190, respectively,
(this strategy is that adopted by the card -minimal repair

ρ 5 ). It would be impor-
tant to exploit well-established information on the data to be repaired in order to
choose the most reasonable repairs among those having minimum cardinality, hence
ranking card -minimal repairs. For instance, in our running example, historical data
retrieved from balance-sheets of past years could be exploited to find conditions
which are likely to hold for the current-year balance-sheet, so that card -minimal
repairs could be ordered according to the number of these conditions which are sat-
isfied in the repaired database. As a matter of fact, consider our running example
in the case that, for all the years preceding 2008, the value of cash sales was never
less than 1000 and the value of receivables was never greater than 200. Then, the
value of cash sales for the current year is not likely to be less than 1000, and the
value of receivables is not likely to be greater than 200. These likely conditions can
be interpreted as weak constraints , in the sense that their satisfaction is not manda-
tory. Weak constraints can be exploited for defining a repairing technique where
inconsistent data are fixed in the “most likely” way. Specifically, the most likely
ways of repairing inconsistent data are those corresponding to card -minimal repairs
satisfying as many weak constraints as possible. For instance, if we consider the
above-mentioned weak constraints in our running example, the card -minimal re-
pair ρ 1 can be reasonably preferred to ρ 5 , since the former yields a database which

satisfies both the weak constraints.
Weak aggregate constraints are aggregate constraints with a “weak” semantics: in
contrast with the traditional “strong” semantics of aggregate constraints (according
to which the repaired data must satisfy all the conditions expressed), weak aggregate
constraints express conditions which reasonably hold in the actual data, although
satisfying them is not mandatory.

Example 4.4. The two conditions defined above (that is, “ it is likely that cash sales
are greater than or equal to 1000 ”, and “ it is likely that receivables are less than or
equal to 200 ”) can be expressed by the following weak aggregate constraints: