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1. See 15 for more on business rules.
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CHAPTER 13 DATA REDUNDANCY AND DATABASE DESIGN: FURTHER THOUGHTS NUMBER ONE
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(The difference is that each exactly in the previous version has been replaced by at most. See 12, Example 6.)
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Identity Decompositions
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While I m on the subject of nonloss decomposition, let me recall something else I said in the previous chapter (in the section Normalization Is Not Enough ): Any relvar can always be nonloss-decomposed, albeit trivially, into what s called the corresponding identity projection. The identity projection of a given relation or relvar is the projection of that relation or relvar on all of its attributes; it s called the identity projection because (obviously enough) it s identically equal to the original relation or relvar. And I added: Of course, it s a bit of a stretch to talk about decomposition in such a situation, because there really isn t any decomposing, as such, going on at all. ADR commented: Your degenerate decomposition, yielding only one projection, fails to merit the appellation decomposition. Even if you feel more sanguine about this state of affairs than I do, the remedy is simple and straightforward the other projection is a variable of type RELATION{}. Well, it seems I do feel more sanguine about this state of affairs than ADR does; I see it as no different in kind from the familiar convention according to which (for example) the SUM, AVG, MAX, and MIN of a single number N are all just N, or the union, join, intersection, and product of a single relvar R are all just R. But it s certainly true that any given relvar R is identically equal to the join of R1 and R2, where: 1. R1 is the projection of R on all of its attributes (i.e., the applicable identity projection). 2. R2 is the projection of R on none of its attributes. R2 here is, as ADR says, of type RELATION{}; it has no attributes at all, and its value is either TABLE_DUM, if R is empty, or TABLE_DEE otherwise.2 I observe, however, that the combination of R1 and R2 isn t a valid nonloss decomposition of R, because it fails to satisfy the requirement that both projections are needed in the reconstruction process. While I m on the subject of identity decompositions, let me remark that any relvar can always be decomposed again trivially, but this time horizontally instead of vertically into the corresponding identity restriction. The identity restriction of a given relation or relvar R is the restriction of R to just those tuples that satisfy the simple boolean expression TRUE, thus:
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2. 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. For further discussion, see, e.g., my book Database in Depth: Relational Theory for Practitioners (O Reilly Media Inc., 2005).
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CHAPTER 13 DATA REDUNDANCY AND DATABASE DESIGN: FURTHER THOUGHTS NUMBER ONE
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R WHERE TRUE It s called the identity restriction because (obviously enough) it s identically equal to the original relation or relvar. I note in passing that a given relation or relvar also always has an empty restriction, which we can denote thus: R WHERE FALSE The (necessarily disjoint!) union of the identity restriction and the empty restriction of a given relation or relvar R is of course identically equal to R.
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Another example I discussed in the previous chapter is repeated in Figure 13-1. Here the parts relvar P has been replaced by two relvars, one ( light parts, LP) containing parts with weight less than or equal to 17 pounds and the other ( heavy parts, HP) containing parts with weight greater than or equal to 17 pounds. The predicates are: LP: Part P# is used in the enterprise, is named PNAME, has color COLOR and weight WEIGHT (which is less than or equal to 17 pounds), and is stored in city CITY. HP: Part P# is used in the enterprise, is named PNAME, has color COLOR and weight WEIGHT (which is greater than or equal to 17 pounds), and is stored in city CITY.
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