Database Reference
In-Depth Information
Chapter 6
Possible Extensions and Open Problems
Abstract We here discuss limits and possible extensions of the framework presented
in the previous chapters for extracting reliable information from numerical data
which are inconsistent w.r.t. a given set of aggregate constraints. Specifically, we
discuss possible refinements of different aspects, involving the form of constraints,
the class of queries, and the minimality semantics.
6.1 Dealing with Different Forms of Constraints
The complexity characterization of RE , MRC , CQA provided in Chapter 3, and of
their variants dealing with steady aggregate constraints, preferred repairs, and range-
consistent answers, is still valid for a form of aggregate constraint more general
than that used so far (Definition 2.1). In fact, all the complexity results still hold
in the presence of aggregate constraints where the conjuncts in the formula
can
be also comparisons between variables and constants, expressed in terms of linear
(in)equalities (such as x i =
φ
1). Observe that, in the case of steady aggregate
constraints, the steadiness restriction on this new form of constraint imposes that
variables occurring in these comparisons cannot be measure variables. This form
enables the definition of constraints such as that imposing, in our Balance Sheet ex-
ample, that the ending cash balance of a year should be the same as the beginning
cash of the next year. It is easy to see that also the MILP-based strategies for com-
puting card -minimal repairs, preferred repairs and range-consistent answers are still
valid in the presence of this new form of constraint.
The form of aggregate constraint can be also changed by modifying the form
of arithmetic expressions specified as attribute expressions. This may result in
deeply changing the expressive power of aggregate constraints. On the one hand,
the expressive power of aggregate constraints does not vary if attribute expres-
sions where addition and scalar multiplication (but not general multiplication) on
numerical attributes are allowed. In fact, this form of constraints can be expressed
by means of our form of aggregate constraints, where attribute expressions con-
x j +
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