By Hervé Gallaire, Jack Minker, Jean Marie Nicolas

Throughout the 1970's a conception of knowledge ases started to evolve seasoned viding a origin for a few current instruments and methods. the importance of this concept is starting to have an effect particu larly with appreciate to the layout and implementation of the relational version of information bases. To additional the state-of-the-art within the concept of knowledge bases, a workshop used to be held in TOulouse, France on December 12-14, 1979. The workshop used to be a sequel to 1 held years formerly and entitled, "Logic and information Base Workshop." As with the sooner paintings store, the conferences have been carried out on the centre d'Etudes et de Recherches de l'Ecole Nationale Superieure de l'Aeronautique et de l'Espace de Toulouse (C.E.R.T). we're happy to recognize the monetary help got from the path des Recherches, Etudes et thoughts d'Armement (D.R.E.T.), and from C.E.R.T. that made the workshop attainable.

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**Sample text**

For if C contains such an fd, then an application of rule FD-JD2 yields the desired fd X + A. That C contains such an fd will be proved in the next section. 0 F F corollary 9 (a) Let C F * [~] fOhd. C be a set of jds and let * [~] • i f and only if C h *[~] be an fohd. Then (b) Let C be a set of fds and jds and let d be an fd or an Then C d if and only if C ~~ d. F Proof: The only if direction is a rephrasing of the completeness theorems. The if direction follows from the soundness of the rules.

In addition, -lJ 1J 1 J 1 X J STX(Q .. ). 1J 1 J 44 BEERI AND VARDI (c) For each i, if for some J', STX(O .. ) is contained in some Q E J X ~ n STX(SJ'» ~. Intuitively, the rule allows us to intersect the X-branches of R with the X-branches of S and connect each of these intersected X-branches to a new x-stem. The new X-stem is not arbitrary. Rather, it must contain the intersection of the appropriate x-stems from ~ and ~ and, in addition, it must contain one of the nonrelativized stems. While condition (b) deals with each component separately, condition (c) deals with the collection as a whole.

All sets of attributes are assumed to be subsets of U. • to denote single attributes and R,S, •••. X,y, ••• to denote sets of attributes. We usually do not distinguish between the attribute A and the set {A}. The union of X and Y is denoted by XY. Thus, ABD denotes the set {A,B,D}. The complement of a set X in U is denoted by x. We will often deal with collections of attribute sets. For clarity we will use the abbreviation asc (attribute set collection) • If ~ = {Rl , ... ,Rk } is an asc then the attribute set of R , denoted ATTR{R), 1S the set U.