Table 1. Changes of Dependencies for Diierent Types of Translation Schemes . . . Dependency Class Fd Eid Egd Tgd Pgd Etgd Projection Fd Eid Egd Etgd Etgd Etgd Join Eid Eid Egd Etgd Etgd Etgd 5 Embedded Implication Dependency

10 Deenition18 (Basically closed dependency class). A class of dependencies D is basically closed if for every translation scheme which consists only of projections and joins, and every D # () D. The following table shows how diierent types of translation schemes change the types of dependencies: From the table we can conclude that the FD's, TGD's and PGD's are not basically closed. On the other hand EID's EGD's and ETGD's are basically closed. Furthermore, we have Theorem 19. The EID's are the smallest basically closed dependency class which contains both the functional and inclusion dependencies. The proof is tedious but straightforward, cf. acknowledgment. The thoerem justiies the choice of EID's as a dependency class, inspite of the undecidability of its consequence problem, CLM81]. For an Entity{Relationship diagram both FD's and IND's are used, so, by stepwise reenements which are dependency preserving, all the EID's have to be available. 6 Conclusions and further research We have introduced the use of translation schemes into database design theory. We have shown how the folklore heuristics of splitting attributes can be made into a precise design tool. 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. This approach can be extended to a full edged design theory for Entity{Relationship design or, equivalently , for database schemes in ER-normal form, cf. MR92]. This approach can also be used to deal with views and view updates, as views are special cases of translation schemes. The latter two applications are currently being developed by the authors and their students and will be included in the full paper. A uniform method for proving lower bounds on the computational complexity of logical theories. 9 Claim 15. At each splitting step of the algorithm there is a translation scheme , s.t. : MOD(new) ! MOD(old), which is onto. Hence, new j = ] (old). This translation scheme is an identity in each place except Y l , where it is vectorized and contains two values from R 2j s : Y 1 l and Y 2 l. Claim 16. Splitting algorithm is a dependency preserving reenement. Proof. The translation scheme is given by the sequential composition of the translation schemes of each step. It now follows by the previous claims and theorem 4 that last j = ] (F). The general case …