Preliminaries - Repairing and Querying Databases under Aggregate Constraints - page 17

form of constraint. For instance, the aggregate constraint introduced in Example 2.6
can be written as

χ () =

30 K .

2.3 Numerical Database Inconsistency

According to the two above-defined forms of constraint on numerical data (domain
and aggregate constraints), we consider two forms of database inconsistency. Specif-
ically, given a database instance D over the database scheme

D , we say that D is in-

consistent w.r.t. D if D contains a tuple t over a relation scheme R ( A 1 : Δ 1 ,..., A n : Δ n )

of D such that, for some A i ∈ M R , it holds that t [ A i ] ∈ Δ i . Moreover, given a set of

aggregate constraints AC on D , we say that D is inconsistent w.r.t. AC if there is

an aggregate constraint ac ∈ AC (of the form (2.1)) such that there is a substitu-

tion θ of variables in x with constants of D making φ ( θ ( x )) true and the inequality

i

1 c i · χ i ( θ (

y i )) ≤ K false on D . We will write D | = AC (resp., D | = D ) to de-

∑

=

note that D is inconsistent w.r.t. AC (resp., w.r.t. D ), and D | = AC (resp., D | = D )

otherwise.

2.4 Steady aggregate constraints

In this section we introduce a restricted form of aggregate constraints, namely steady
aggregate constraints . On the one hand, steady aggregate constraints are less ex-
pressive than (general) aggregate constraints, but, on the other hand, we will show
in Chapter 4 that computing a card -minimal repair w.r.t. a set of steady aggregate
constraints can be accomplished by solving an instance of a Mixed Integer Linear
Programming (MILP) problem [33]. This allows us to adopt standard techniques ad-
dressing MILP problems to accomplish the computation of a card -minimal repair.
It is worth noting that the loss in expressiveness is not dramatic, as steady aggre-
gate constraints suffice to express relevant integrity constraints in many real-life
scenarios. For instance, all the aggregate constraints introduced in “Balance Sheet”
example can be expressed by means of steady aggregate constraints.
Let R

( A 1 ,..., A n ) be a relation scheme and R ( x 1 ,..., x n ) an atom, where each

x j is either a variable or a constant. For each j , we say that the term x j is

associated with the attribute A j . Moreover, we say that a variable x i is a measure
variable if it is associated with a measure attribute.

∈ [ 1 .. n ]

Definition 2.2 (Steady aggregate constraint). An aggregate constraint ac on a
given database scheme D is said to be steady if the following conditions hold:

1. for every aggregation function R , e ,α on the right-hand side of ac , no measure

;
2. measure variables occur at most once in ac ;

attribute occurs in α