To close this chapter, I’d like to return to the example I opened it with. Just to remind you, that example involved the nonloss decomposition of our usual suppliers relvar S into its projections SNC (on {SNO,SNAME,CITY}) and ST (on {SNO,STATUS}). That decomposition lost the FD {CITY} → {STATUS}, with the consequence that updates to either of the projections sometimes required updates to the other, in order to enforce the constraint that each city has just one status. By contrast, the “sensible” decomposition into projections SNC (on {SNO,SNAME,CITY}) and CT (on {CITY,STATUS}) suffers from no such problem—updates can be made to either projection without regard to the other.[64]
For the sake of the present discussion, let me refer to decompositions like the one into SNC and ST as bad and decompositions like the one into SNC and CT as good. As we’ve seen, then, the projections in a good decomposition can be updated independently of each other; for that reason, they’re sometimes referred to explicitly as independent projections. By contrast, the projections in a bad decomposition aren’t independent in that same sense. So we can say that in order to preserve FDs, we want a decomposition in which the projections are independent. And there’s a theorem, due to Jorma Rissanen, that can help in this regard. Before I state that theorem, however, let me give a precise definition of what it means for two projections to be independent:
Definition: Projections R1 and R2 of relvar R are independent if and only if every FD that holds in R also holds in the join of R1 and R2.
Here now is the theorem:
Rissanen’s Theorem: Let relvar R, with heading H, have projections R1 and R2, with headings H1 and H2, respectively; further, let H1 and H2 both be proper subsets of H, let their union be equal to H, and let their intersection not be empty.[65] Then projections R1 and R2 are independent if and only if (a) their common attributes constitute a superkey for at least one of them and (b) every FD that holds in R is implied by those that hold in at least one of them.
Consider the “good” decomposition of S into its projections SNC and CT. Those two projections are independent, because (a) the set of common attributes is just {CITY} and {CITY} is a (super)key for CT, and (b) every FD that holds in S either holds in one of the two projections or is implied by those that do (see the next chapter). By contrast, consider the “bad” decomposition into the projections SNC and ST. Here the projections aren’t independent, because the FD {CITY} → {STATUS} can’t be inferred from those holding in those projections—although it’s at least true that the set of common attributes, {SNO}, is a key for both.
As a historical note, it was Rissanen’s work on independent projections (which was done, or at least published, in 1977) that laid the foundation for the theory of what we now call FD preservation.