The
hierarchy schema does not provide semantics to discard any of such structures.
Dimension constraints are Boolean combinations of atomic statements called atoms.
We use the standard Boolean connectives ( ??§ ,???, ?¬, ?‡”, ?‡’) . Atoms are the expressions
in brackets. As an example, the constraint:
?¬ ??©Product,ElectricalCategory??? ??? ?¬ ??©Product,MusicCategory???
states that the products have parents in either ElectricalCategory or MusicCategory
but not in both of them.
We need negation, conjunction, and disjunction, to restrict structures and to reason
about summarizability (Hurtado et al., 2005). This motivated us to incorporate the
entire expressiveness of the Boolean connectives into dimension constraints. Dimension
constraints also incorporate atoms of the form ??©Product, Brand= b3??? called
equality atom which make it possible to place restrictions conditioned on particular
elements. As an additional example, we may write that the products that belong to
brand b3 are sold on shelves, using the following constraint:
??©Product, Brand=b3??? ?‡’ ??©Product,Shelf???.
The constraints we have already showed are statements about the applicability of
categories. We may also need to place restrictions about the applicability of hierarchy
paths. We use paths atoms for this purpose. As an example, in the dimension
of Figure 7, we may state that the products that belong to the electrical category
ec2 have the path Brand??°Category??°Department??°All in their structure, using the
following constraint:
??©Product, ElectricalCategory=ec2??? ?‡’ ??©Product,Brand,Category,Department,All???.
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