Advances in Data Base Theory: Volume 2 by Joachim Biskup, Hans Hermann Brüggemann (auth.), Hervé

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've got edited. every one booklet has been the outgrowth of papers held at a workshop in Toulouse, France. the 1st workshop, held in 1977 centred totally on the real subject of good judgment and databases. The e-book, common sense and Databases was once the results of this attempt. the varied makes use of of common sense for databases corresponding to its use as a theoretical foundation for databases, for deduction and for integ­ rity constraints formula and checking was once defined within the chapters of the e-book. The curiosity generated by means of the 1st workshop ended in the deci­ sion to behavior different workshops all in favour of theoretical concerns in databases. as well as good judgment and databases the kinds of papers have been extended to incorporate different very important theoretical matters akin to dependency idea which, even though it occasionally makes use of common sense as a foundation, doesn't healthy with our meant that means of common sense and databases explored on the first workshop. as a result of the broader assurance, and since we expected extra workshops, the second one e-book used to be entitled, Advances in Database conception - quantity 1. The ebook "Logic and Databases" will be thought of quantity zero of this series.

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Finally, if the subhypergraph it = of H. where N = eEEYU E" e and E E' U E". is not connected then the connectivity may be obtained by adding to t one and only one edge from E. Hence all and only the subhypergraphs H = of H such that: U eE E' U E" e U e . E= E' U E" U {e} and are the nonredundant coverings over N. is proved. Since e E IE"' I 2. IEI EIII the theorem AUSIELLO, D'ATRI AND MOSCARINI 40 N C e, U < eEE3 E' U e, E3 > eEE3 {e 7 ,e 8 ,e 9 } E" {ell,e12,elO,e14} E {e l ,e 2 } < E'" E < < U eEE l U eEE 2 Figure 10.

To prove the only if part we show that if H is y-acyclic we may establish a 1-1 correspondence between the edges of two distinct nonredundant coverings over any set N. In fact, let HI, H" be two such coverings and let ei be any edge in HI and not in H". We show how we can associate with ei a unique edge ek which is in H" and not in HI, and we show that such an association is a 1-1 mapping. n,m Let be two nodes in N such that ei belongs to a nonredundant I ••• ,eq,> I f rom nom - t -"~n HI Since H" i s a cover~ng " pa th p I = in H" from to iii where ql = q".

Furthermore to determine both EI(e') and f(e') requires polynomial time in lEI + INI. REFERENCES 1. Aho, A. , Sagiv, Y. and Ullman, J. D. [1979] "Equivalence of relational expressions," SIAM Journal of Computing 8(2) (May 1979) 218-246. 2. , Mendelzon, A. , Ullman, J. , and Yannakakis, M. [198la] "Properties of acyclic da tabase schemes," Proceedi ngs 13th Annual ACM Symposi um on Theory of Computing, Milwaukee, Wisconsin (May 1981) 355-362. 3. , Maier, D. and Yannakakis, M. [198lb] "On the desirability of acyclic database schemes", IBM Report RJ 3131, San Jose, California (May 1981) to appear in Journal of the ACM.

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