By Joachim Biskup, Hans Hermann Brüggemann (auth.), Hervé Gallaire, Jack Minker, Jean Marie Nicolas (eds.)

This is the 3rd booklet dedicated to theoretical concerns in facts­ bases that we have got edited. every one publication has been the outgrowth of papers held at a workshop in Toulouse, France. the 1st workshop, held in 1977 concentrated totally on the real subject of good judgment and databases. The ebook, good judgment and Databases was once the results of this attempt. the various makes use of of good judgment for databases comparable to its use as a theoretical foundation for databases, for deduction and for integ­ rity constraints formula and checking used to be defined within the chapters of the booklet. The curiosity generated by means of the 1st workshop resulted in the deci­ sion to behavior different workshops curious about theoretical concerns in databases. as well as common sense and databases the kinds of papers have been multiplied to incorporate different very important theoretical matters comparable to dependency concept which, even though it occasionally makes use of good judgment as a foundation, doesn't healthy with our meant which means of good judgment and databases explored on the first workshop. as a result of broader assurance, and since we expected extra workshops, the second one ebook was once entitled, Advances in Database thought - quantity 1. The publication "Logic and Databases" might be thought of quantity zero of this series.

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In this caseJe should have the same intersec~lon~ as ei and the same intersecti~ns as e! in H'; furthermore, if N. ) denote the set of nodes in N J J. belonging to ei (ej) we shoula have that N. ~ ek and Nj ~ e Since H' is nonredundant at least one of the fol16wing condit10ns lias to hold: k k. there exists an edge e ' in H' such that e~ 1 () e ' # e~ J n e' , or In both cases we musr have two nodes in ek connected by a path (containing ei and ej) with length greater than or equal to two. Hence, fact (c) and the theorem are proved.

1983] "Windows on the world", Proceedings Annual Meeting of SIGMOD, SIGMOD Record 13(4) (May 1983) 68-78. 19. Maier, D. and Ullman, J. D. [1982] "Connections in acyclic hypergraphs", Proceedings First SIGACT-SIGMOD Symposium on Principles of Database Systems, Los Angeles, California (March 1982) 34-39. 20. Osborn, S. L. [1979] "Towards a universal relation interface", Proceedings 5th Conference on Very Large Data Bases, Rio de Janeiro, Brazil (October 1979) 52-60. MINIMAL COVERINGS OF ACYCLIC SCHEMATA 51 21.

For this purpose we have to verify the assertion of the following theorem. Theorem 6. 5. Then N is the (up to renaming unique) reduction result eof G • e (2) In particular, the final value of N is the (up to renaming) unique reduction result of the output value of G. 4. e+l: We have to study the following situation: N is the reduction result of G by the induction e e hypothesis; k ~ Ge is the new name with l1t C w(Ne ), cf. 3 during the performance of the eth loop. BISKUP AND BRtlGGEMANN 18 Ge + l = Ge U {k}; Ne + l is the reduction result of Ne U {k}; * G e add k ) G =G U {k}- e+l e ) -* - - ...

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