Databases Reference
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At the conclusion of this procedure, the elements of S are the headings of a set of BCNF relvars into which R
can be nonloss decomposed, though not necessarily without losing FDs. Also, let me point out for the record that
the procedure makes no mention of either 2NF or 3NF.
It's a bit of a digression from the main theme of this chapter, but I'd like to elaborate briefly on the concept of an
identity projection. Here's a definition (I define it for relvars, but of course an analogous concept applies to
relations as well):
Definition: The identity projection of a given relvar is the projection of that relvar on all of its attributes.
Now, it should be obvious that any relvar can be nonloss decomposed, albeit trivially, into its identity
projection. However, some people don't like to think of such a decomposition as being a decomposition, as such, at
all (as I said in connection with the SJT example, “there isn't much decomposition, as such, going on here”). If you
happen to be one of those people, then, you might prefer the following way of looking at the matter. Let relvar R
have heading H . Then it's certainly true that the FD {} → {} holds in R , where {} is the empty set of attributes (this
FD is trivial, of course, and holds in every relvar). By Heath's Theorem, therefore—take X , Y , and Z to be {}, {},
and H , respectively— R can be nonloss decomposed into its projections R1 and R2 , where:
The heading XY of R1 is the union of {} and {}, which reduces to just {}; i.e., R1 is the projection of R on no
attributes at all, and its value is either TABLE_DUM, if R is empty, or TABLE_DEE otherwise. 11
The heading XZ of R2 is the union of {} and H , which reduces to just H ; i.e., R2 is the projection of R on all
of its attributes, or in other words the identity projection of R .
I hope it's clear that this decomposition is nonloss— R is certainly equal to the join of R1 and R2 . (On the other
hand, it's also true that the combination of R1 and R2 fails to meet the usual requirement that both projections
should be needed in the reconstruction process.)
While I'm on the subject of what might be called identity decompositions , let me remark that any relvar can
also always be decomposed (again trivially, but this time “horizontally” instead of “vertically”) into the
corresponding identity restriction . 12 Here's a definition (again I define it for relvars, but of course an analogous
concept applies to relations as well):
Definition: The identity restriction of a given relvar R is any restriction of R in which the restriction
condition is identically true—in other words, any restriction of R that's logically equivalent to one of the
following form:
11 TABLE_DUM and TABLE_DEE are pet names for, respectively, the unique relation with no attributes and no tuples and the unique relation
with no attributes and one tuple. (We've met these relations before, in the answer to Exercise 2.8 in Appendix D.) For further discussion, see
SQL and Relational Theory .
12 For precise definitions of restriction and the associated notion of a restriction condition , see the answer to Exercise 13.13 in Appendix D.
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