Cylindric Algebras and Relational Databases

The success of the relational data model introduced by Codd [Cod,70] can — at least partly — be attributed to its clarity and succinctness. The semantics of the model are constraints among and within the base relations which, as it turned out, could be formulated in various fragments of first order logic. Therefore, a natural approach seems to be the general framework of cylindric structures. The two main examples of these are concrete algebras of

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-ary relations on the one hand, and algebras of first order sentences on the other. It is interesting to note that throughout most of the research on the theory of relational databases, this dichotomy can be observed, albeit in a different terminology. Codd’s relational algebra and domain- or tuple calculus are well known examples for this phenomenon, and, keeping in mind the connection between cylindric algebras and first order logic, it comes as no surprise that the resulting query languages are of the same expressive power. The extended relations of [Yan-Pap,82] are, loosely speaking, nothing else than an embedding of some finite cylindric algebra into an infinite algebra of formulae, and a transformation from finite dimensional relations to formulae (and infinite dimensional cylindric algebras) is implicitly present in [Cos,87].