8.4 Er-schemes

A uniform method for proving lower bounds on the computational complexity of logical theories. If we restrict our translation schemes to compositions of projections and joins and our dependencies to EID's, the previous section showed that for X = A, B or C, (X) and (X-EID) are equivalent, because the EID's are basically closed, cf. theorem 21. This is the appropriate framework to compare transformations of ER-schemes and to address the Fundamental Problem of ER{Database Design. We plan to address this issue in our future work. We have introduced the use of translation schemes into database design theory. We have shown how they capture disparate notions such as information preservation and dependency preservation in a uniform way. We have shown how they relate to normal form theory and have stated what we think to be the Fundamental Problem of Database Design. Several resulting research problems have been explicitly stated in the paper. We have shown that the Embedded Implicational Dependencies are all needed, when we deal with stepwise reenements of database schemes speciied by Functional and Inclusion Dependencies. As the material presented grew slowly while teaching database theory, its foundational and didactic merits should not be underestimated. Over the years our students of the advanced database theory course connrmed our view that traditional database design lacks coherence and that this approach makes many issues accessible to deeper understanding. Our approach via dependency preserving translation{reenements can be extended to a full edged design theory for Entity{Relationship design or, equivalently , for database schemes in ER-normal form, cf. MR92]. It is also the appropriate framework to compare transformations of ER-schemes and to address the Fundamental Problem of ER{Database Design. Translation schemes can also be used to deal with views and view updates, as views are special cases of translation schemes. The theory of complementary views from BS81] can be rephrased elegantly in this framework. It is connected with the notion of translation schemes invariant under a relation and implicit de-nability, Kol90]. Order invariant translation schemes play an important role in descriptive complexity theory, Daw93] and Mak94]. The theory of independent complementary views of KU84] exhibits some severe limitations on the applicability of BS81]. In spite of these limitations it seems worthwhile to explore the connection between independent views and transformation invariant for certain relations further. The latter two applications are currently being developed by the authors and their students and will be …