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Database Management Systems/Managing Database

Notes 2. The processes are described using different frames of reference, but there exists a “common”
frame of reference (which is a refinement of both of these). In this case, p is a specialization
1
of p if and only if the refinement of p is a specialization of the refinement of p under the
0 1 0
common frame of reference. Thus this second case is reduced to the first by means of
refinement.
We propose that one useful way to operationalize this notion of specialization is in terms of a set
of transformations for any particular process representation, which, when applied to a process
description, produce a description of a specialization of that process. The two part definition of
specialization suggests that two sorts of transformations will be needed:

A specializing transformation is an operation which, when applied to a process described using
a given representation and a given frame of reference, results in a new process description under
that representation and frame of reference corresponding to a specialization of the original
process. Specializing transformations change the extension of a process while preserving the
frame of reference.
In contrast, a refining transformation is an operation which changes the frame of reference of a
process while preserving its extension, producing a process description of the same process
under a different frame of reference.
For each type of transformation there is a related inverse type: a generalizing transformation
acts on a process description to produce a generalization of the original process and is thus the
inverse of a specializing transformation. Similarly, an abstracting transformation is the inverse
of the refining transformation, producing a new description of the same process under a frame
of reference for which the original frame is a refinement.
Given that the refining/abstracting transformations preserve the extension of a process, it follows
from our definition of process specialization that a specializing transformation composed with
refining/abstracting transformations in any sequence produces a specialization. The analogous
statement holds for generalizing transformations.
A set of refining/abstracting transformations is said to be complete if for any process p described
under a frame of reference, the description of that process under any other frame of reference can
be obtained by applying to p a finite number of transformations drawn from the set.
A set of specializing transformations is said to be locally complete if for any frame of reference
and any process p described using that frame of reference, then any specialization of p described
under that frame of reference can be obtained by applying to p a finite number of transformations
drawn from the set. This corresponds to the first part of the definition of process specialization
given above.

There is also a notion of completeness corresponding to the second part of the definition. A set
of specializing transformations and refining/abstracting transformations is said to be globally
complete if for any process p, any specialization of p for which a common frame of reference
exists can be obtained by applying to p a finite number of transformations drawn from the set.
Proposition: Let A be a complete set of refining/abstracting transformations and S be a locally
complete set of specializing transformations. Then A [[union]] S is globally complete.
Proof: Consider a process p and a specialization p for which a common frame of reference
0 1
exists. Since A is complete, one can apply a finite sequence of transformations from A to p to
0
produce its refinement in the common frame of reference. By local completeness one can then
apply specializing transformations to produce the refinement of p (since it is a specialization of
1
the refinement of p by assumption). Finally, by the completeness of A one can transform the
0
refinement of p into p .
1 1

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