Database Reference
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schema mappings is that they cannot be obtained by the first-order logic formulae.
From the logical point of view, we can consider the representation of inter-schema
mappings and their compositions by SOtgds also in our approach, but with different
meaning assigning to them.
In our case, the two equal compositions are not necessarily logically equivalent
formulae (which requires the consideration of all intermediate databases involved
in this composition). We will use SOtgds to define a mathematical framework from
which we will be able to determine the strict meaning of composed mappings at an
instance-level, as a quantity of information (set of views) specified by mapping to
be transmitted from the active domain of the given source instance into the target
instance.
Moreover, in order to pass to a categorical logic and its functorial semantics, we
will use an algebraic representation based on the theory of operads both at schema
(e.e., database sketches) and at instance database levels. We will show that this al-
gebraic representation based on operads is semantically equivalent to the represen-
tation of the logical mappings based on SOtgds. Consequently, we will develop the
algorithms for the transformation of SOtgds into the mapping operads and the algo-
rithms for determination of the strict semantics at an instance database level. We will
show that the egds can be represented as particular SOtgds and mapping operads so
that the whole set of integrity constraints over a given schema can be equivalently
represented by a schema mapping as well. The algorithm for composition of SOtgds
givenin[
5
] for the data-exchange setting has to be generalized for this more general
semantics of schema mappings that we intend to use.
2.2
Transformation of Schema Integrity Constraints into
SOtgds
Codd initially defined two sets of constraints but, in his second version of the rela-
tional model, he came up with the following integrity constraints:
•
Entity integrity
. The entity integrity constraint states that no primary key value
can be null. This is because the primary key value is used to identify individual
tuples in a relation. Having null value for the primary key (PK) implies that
we cannot identify some tuples. This also specifies that there may not be any
duplicate entries in the primary key column key row.
•
Referential Integrity
. The referential integrity constraint is specified between two
relations and is used to maintain the consistency among tuples in the two re-
lations. Informally, the referential integrity constraint states that a tuple in one
relation that refers to another relation must refer to an existing tuple in that rela-
tion. It is a rule that maintains consistency among the rows of the two relations.
•
Domain Integrity
. The domain integrity states that every element from a relation
should respect the type and restrictions of its corresponding attribute. A type can
have variable length which needs to be respected. Restrictions could be the range
of values that the element can have, the default value if none is provided, and if
the element can be NULL.
