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CHAPTER 19 THERE S ONLY ONE RELATIONAL MODEL
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By the way, in case it s not obvious, I should add that today s O/R DBMSs are not true O/R DBMSs, because they re not true relational DBMSs. In other words, they re not built on what I claim is the only sensible O/R model.
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Another reader wrote to Fabian Pascal, editor of the website http://www.dbdebunk.com, to ask: How does [the fact that the storage level in a relational system is not itself relational square] with the fundamental principle of the relational model All information in the database must be cast explicitly in terms of values in relations and in no other way In his reply, Fabian said in part: [You are confusing] the logical and physical levels ... The relational model is purely logical and has absolutely nothing to do with storage! This response is 100 percent correct, of course. Unfortunately, Fabian went on to say this: What Date means [here Fabian is referring to some writings of my own on the same subject] is that any atomic values can be represented by a relational DBMS, no matter how complex the representation (e.g., domains can be relation-based). Gentle reader, please delete the word atomic here! It has no precise definition. What I do claim (and have claimed ever since about 1992, when I first realized that to talk in terms of this fuzzy atomicity concept was misleading and counterproductive) is that if A is some attribute of some relation, then A can be defined in terms of absolutely any type (or domain, if you prefer) whatsoever. So you can have attributes whose values are integers, attributes whose values are strings, attributes whose values are polygons, attributes whose values are arrays, attributes whose values are relations, ... and on and on. Of course, it s crucial (as Fabian suggests) that we make a distinction between values of any given type, on the one hand, and the representation of those values under the covers, on the other but the idea that these two concepts should be kept rigidly apart isn t one that s peculiar to the relational world.
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The last letter I want to discuss didn t include any relational misconceptions as such; however, it still makes sense to deal with it here. I ll refer to the writer as CC. CC asks some questions regarding the terminology of relations and relationships and related matters. To be specific, he or she asks whether relations in relational database theory are equivalent to relations in set theory (quotation marks as in the original letter). I replied as follows. This question requires a rather lengthy answer! I think the overriding point is that the relation construct as understood in relational database theory is based on, but is not identical to, its mathematical counterpart. Here are some of the principal points of difference between the two: RDB relations are typed they emphasize, much more than mathematical relations do, the significance of the relation heading. Note: Here and throughout the remainder of these remarks I use RDB as a convenient abbreviation for relational database. An explanation of the term relation heading can be found in many places (see, e.g., An Introduction to Database Systems).
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CHAPTER 19 THERE S ONLY ONE RELATIONAL MODEL
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The heading of an RDB relation has no left-to-right ordering to its attributes; the heading o f a mathematical relation does have such an ordering. (In fact, of course, the term attribute is never used in the RDB sense at all in a mathematical context.) In particular, RDB relations have named attributes, while mathematical relations don t. Those names in turn play a crucial role in the relational algebra, especially with respect to relation type inference. And to spell the point out, those attribute names are conceptually distinct from the underlying domain or type names. The primary emphasis in mathematics is on binary relations specifically. RDB relations, by contrast, are n-ary, where n can be 0, 1, 2, 3, ... Note the cases n = 0 and n = 1 in particular! The case n = 0 turns out to be crucially important for all kinds of fundamental reasons. Relational algebra: Very little work seems to have been done in mathematics or logic on general n-ary relational operators, presumably because of the emphasis already noted on binary relations specifically. For example, Patrick Suppes s book Introduction to Logic (Van Nostrand, 1957) defines an operator called relative product: If r(A,B) and s(B,C) are two binary relations, then their relative product t(A,C) is the binary relation consisting of all pairs (a,c) such that, for some b, the pair (a,b) appears in r and the pair (b,c) appears in s. In RDB terms, this is the projection over (A,C) of the join of r and s over B.9 But notice how the operation is specifically defined to produce a binary relation as its result; the ternary relation that is the intermediate result the join is never explicitly mentioned. Thus, although operators such as join (and all of the other operators of the relational algebra) are clearly applicable to mathematical relations, it s fair to say that they were first defined (for the most part) in the context of relations in the RDB sense. Indeed, the theory of such operators (including the laws of expression transformation, the associated principles of optimization, etc.) can reasonably be regarded as a new branch of mathematics, and the theory in question is one that arose specifically as part of the development of the relational approach to the problem of database management. Dependency theory: To say it again, the emphasis in mathematics tends to be on binary relations specifically, whereas the emphasis in the relational model is on n-ary relations instead. The entire field of what is now usually called dependency theory the large body of theorems and techniques concerning such matters as functional dependence, multivalued dependence, join dependence, higher normal forms, candidate keys, etc. is crucially dependent on this difference in emphasis. For example, the concept of Boyce/Codd normal form (BCNF) is relevant only to relations of degree three or more, because all (well, almost all) relations of degree less than three are necessarily in BCNF. In fact, the entire field of dependency theory like the theory of n-ary relational operators already mentioned above can be regarded as a new branch of mathematics, one that was brought into being by the special requirements of a theory of data and a theory of n-ary relations (as opposed to a theory of binary relations merely).
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9. In fact, it s the relational compose operator (see, e.g., the Third Manifesto book).
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