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clause of
an aggregate query means considering queries which satisfy a “steadiness” condi-
tion analogous to that imposed on steady aggregate constraints (see Definition 2.2).
Given an instance
D
of
Basically, the restriction that no measure attribute occurs in the
WHERE
D
, the evaluation of an aggregate query
q
on
D
will be
denoted as
q
(
D
)
.
Example 5.2.
Queries
q
1
,
q
2
and
q
3
defined in Example 5.1 can be expressed as fol-
lows:
q
1
=
SELECT MAX
(
Value
)
FROM
BalanceSheets
Subsection
=
'cash sales'
WHERE
q
2
=
SELECT MIN
(
Value
)
FROM
BalanceSheets
Subsection
=
'cash sales'
WHERE
q
3
=
SELECT SUM
(
Value
)
FROM
BalanceSheets
Subsection
=
'cash sales'
WHERE
We now introduce the fundamental notion of range-consistent answer of an ag-
gregate query. Basically, it consists in the narrowest range
[greatest-lower bound
(glb), least-upper bound (lub)]
containing all the answers resulting from evaluat-
ing the query on every database resulting from the application of a
card
-minimal
repair
1
.
Definition 5.2 (Range-consistent query answer).
Let
D
be a database scheme,
AC
a set of aggregate constraints on
D
,
q
an aggregate query on
D
, and
D
an instance of
D
. The
range-consistent query answer
of
q
on
D
, denoted as
CQA
q
D,AC
(
D
)
is the empty interval 0, in the case that
D
admits no repair w.r.t.
AC
[
,
]
, otherwise, where:
i) for each
card
-minimal repair
, or the interval
glb
lub
ρ
for
D
w.r.t.
AC
, it holds that
glb
≤
q
(
ρ
(
D
))
≤
lub
;
ii) there is a pair
ρ
,
ρ
of
card
-minimal repairs for
D
w.r.t.
(
ρ
(
AC
such that
q
D
)) =
(
ρ
(
glb
and
q
D
)) =
lub
.
Example 5.3.
In our running example, the narrowest range including the evaluations
of query
q
1
on every database resulting from the application of a
card
-minimal re-
pair is
[
1110
,
1150
]
(as shown in Example 2.12, the
card
-minimal repairs are
ρ
1
and
ρ
5
;
q
1
evaluates to 1150 and 1110 on the databases repaired by
ρ
1
and
ρ
5
, respec-
tively). Hence, the range-CQA of query
q
1
is
[
1110
,
1150
]
. Similarly, it is easy to
see that the range-CQAs of
q
2
and
q
3
are
[
900
,
1110
]
and
[
2010
,
2260
]
, respectively.
5.3 Evaluating Range-Consistent Query Answer
In this section, we define a strategy for computing range-consistent answers of ag-
gregate queries in the presence of steady aggregate constraints. Before describing
1
It is worth
n
oting that, as in this chapter we assume that the
do
main of measure attributes is
bounded by
M
,
card
-minimal repairs considered here consists of
M
-bounded atomic updates.
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