Introduction - Repairing and Querying Databases under Aggregate Constraints - page 4

As shown in the following example, when tuples contain both correct and erro-
neous components the two repair semantics discussed above do not coincide.

Example 1.2. Consider the following database scheme consisting of the relation
Employee

( Code , Name , Salary ) , where the attribute Code is a key for the relation.

Assume that the constraint ∀ x , y , z ¬ [ Employee ( x , y , z ) ∧ z < 10000 ] is defined, stat-

ing that each employee must have salary greater than 10000.
For the following (inconsistent) instance of the relation Employee , under the
repair semantics of deletion/insertion of tuples, there is a unique repair: the empty
database instance.

C ode N a m e S alary
111 John 1000

On the other hand, if the repaired database is obtained by changing attribute val-
ues, there are infinitely many repairs, each of them containing a tuple of the form

, where c a constant greater than or equal to 10000.
Thus, under the latter repair notion, the consistent answer to the query asking
for the existence of the employee with code '111' is yes , whereas under the former
repair notion the consistent answer is no . This happens because when we delete a
tuple containing a wrong component, we also lose the correct components as an
undesirable side effect.

111 , John , c

Several theoretical issues regarding the consistent query answers problem have
been widely investigated in literature and some techniques for evaluating consistent
answers have been proposed too. The problem of computing consistent answers has
been studied among several dimensions, such as the repair semantics, the classes
of queries and constraints. Many approaches in literature assume that tuple inser-
tions and deletions are the basic primitives for repairing inconsistent data. More
recently, repairs consisting also of value-update operations have been considered.
The complexity of computing consistent answers for different classes of first-order
queries and aggregate queries has been investigated in presence of several classes of
integrity constraints.
In the following section, we introduce the problem of repairing and querying
numerical databases violating a particular class of integrity constraints (called ag-
gregate constraints ), whose investigation is the main topic of this manuscript.

1.2 Numerical Data and Integrity Constraints

Researchers have deeply investigated several issues related to the use of integrity
constraints in relational databases. A great deal of attention has been devoted to the
problem of extracting reliable information from databases containing pieces of in-
formation inconsistent w.r.t. some integrity constraints. Most of the previous work
in this area deals with “classical” forms of constraint (such as functional and in-
clusion dependencies), and proposes different strategies for updating inconsistent