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CHAPTER 13 DATA REDUNDANCY AND DATABASE DESIGN: FURTHER THOUGHTS NUMBER ONE
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also that (a) each class has just one associated status, and (b) each city has just one associated status as well, but (c) classes and cities are otherwise quite independent of each other. Then the relvar satisfies these two functional dependencies (FDs): { CLASS } { STATUS } { CITY } { STATUS } (Perhaps this is an example of one of the pathological cases that ADR refers to in his comment.) Anyway, I ll leave the details as an exercise, but it should be apparent that the following is a valid nonloss decomposition for this revised suppliers relvar (I ignore attribute SNAME for simplicity): SCC { S#, CLASS, CITY } KEY { S# } CLS { CLASS, STATUS } KEY { CLASS } CTS { CITY, STATUS } KEY { CITY } Observe in particular that attribute STATUS appears here as a nonkey attribute in more than one projection, and the decomposition thus violates ADR s proposed rule. Yet, to repeat, the decomposition is surely valid, and the proposed rule is thus not quite right. What s more, it seems to me that the more difficult it is to get it right, the more reasonable it is to regard the correct version, if and when we find it, as indeed another principle.
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I made the claim in 12 that functional dependencies were a special case of join dependencies.4 But ADR wrote: I find this claim very misleading ... I believe such claims ... make 4NF and 5NF more difficult to understand than they should be ... You only have to look at a functional dependency (FD) and a join dependency (JD) to see that an FD isn t a JD. However, we can indeed derive from an FD the projections of a binary JD. (I tried to write down the derivation rule but realised that it s not that trivial and I would be likely to make a mistake.) These comments gave me furiously to think! I was very surprised at the suggestion that an FD might not be a JD, and had to go back and revisit some very basic concepts in order to convince myself that my claim that every FD is a JD was indeed correct. Let me elaborate. Of course, it s true that we typically use different formalisms different syntax, in other words to express FDs and JDs. For example, let s go back to the original suppliers relvar S:
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4. I assume you re familiar with these concepts. Definitions are given in the previous chapter; more detailed explanations can be found in many places see, e.g., the O Reilly book mentioned earlier (Database in Depth: Relational Theory for Practitioners).
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CHAPTER 13 DATA REDUNDANCY AND DATABASE DESIGN: FURTHER THOUGHTS NUMBER ONE
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S { S#, SNAME, STATUS, CITY } KEY { S# } As we know, this relvar can be nonloss-decomposed into projections as follows: SN { S#, SNAME } KEY { S# } ST { S#, STATUS } KEY { S# } SC { S#, CITY } KEY { S# } Thus, we can say that relvar S satisfies the following FDs: { S# } { SNAME } { S# } { STATUS } { S# } { CITY } We can also say it satisfies the following JD: { { S#, SNAME }, { S#, STATUS }, { S#, CITY } } And these two assertions are clearly different in form but they are logically equivalent. In fact, there s a theorem (Heath s theorem) that effectively says as much: Let A, B, and C be subsets of the heading of relvar R such that the (set-theoretic) union of A, B, and C is equal to that heading. Let AB denote the (set-theoretic) union of A and B, and similarly for AC. If R satisfies the FD A B, then R is equal to the join of its projections on AB and AC. Observe that the third (final) sentence here can equivalently be written: If R satisfies the FD A B, then R satisfies the JD {AB,AC}.
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Observe too that this same sentence addresses the last part of ADR s comment (regarding the rule by which we can derive the projections of a binary JD from a given FD). Incidentally the following point is a little tangential to the main topic under discussion, but I think it s interesting there s a very pleasing parallelism (first noted by Ron Fagin) among the definitions of: Boyce/Codd normal form, BCNF, which is the normal form so long as we limit our attention to functional dependencies only Fourth normal form, 4NF, which is the normal form when we take multi-valued dependencies, MVDs, into account as well Fifth normal form, 5NF, which is the normal form when we take general join dependencies into account as well
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