Symbol and connectives and then the selection of r

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Unformatted text preview: ute domains), equality symbol =, negation symbol :, and connectives _ and ^. Then, the selection of R by F , denoted _ F (R), is a paraconsistent relation on scheme , given by _ F (R)+ = where F F (R + ); _ F (R)? = R? :F ( ( )); is the usual selection of tuples satisfying F from ordinary relations. 2 Proposition 6 The operators _ and _ are strong generalisations of and , respectively. Proof Similar to that of Proposition 5. 2 Example 1 Strictly speaking, relation schemes are sets of attribute names, but in this example we treat them as ordered sequences of attribute names, so tuples can be viewed as the usual lists of values. Let fa; b; cg be a common domain for all attribute names, and let R and S be the following paraconsistent relations on schemes hX; Y i and hY; Z i, respectively: R+ = f(b; b); (b; c)g; S + = f(a; c); (c; a)g; R? = f(a; a); (a; b); (a; c)g; S ? = f(c; b)g: _ Then, R 1 S is the following paraconsistent relation on scheme hX; Y; Z i: _ (R 1 S )+ = f(b; c; a)g; _ (R 1 S )? = f(a; a; a); (a; a; b); (a; a; c); (a; b; a); (a; b; b); (a; b; c); (a; c; a); (a; c; b); (a; c; c); (b; c; b); (c; c; b)g: _ Observe how (R 1 S )? blows up to contain extensions of all tuples in R? and S ?. Now, _ _ hX;Z i (R 1 S ) becomes the following paraconsistent relation on scheme hX; Z i: _ _ hX;Z i (R 1 S )+ = f(b; a)g; _ _ hX;Z i (R 1 S )? = f(a; a); (a; b); (a; c)g: The tuples in the negative component of the projected paraconsistent relation are such that all their extensions were present in the negative component of the original para_ consistent relation. Finally, _ :X =Z ( _ hX;Z i (R 1 S )) becomes the following paraconsistent relation on the same scheme: _ _ :X =Z ( _ hX;Z i (R 1 S ))+ = f(b; a)g; _ _ :X =Z ( _ hX;Z i (R 1 S ))? = f(a; a); (a; b); (a; c); (b; b); (c; c)g: All tuples that do not satisfy the selection condition always make it to the negative component of the selected paraconsistent relation. 2 11 4 An Application In this section we give an application of the algebra on paraconsistent relations. We brie y present a bottom-up method for constructing the weak well-founded model of any general deductive database. For a somewhat detailed exposition on general deductive databases the reader is referred to 17], and on the weak well-founded model to 9]. General Deductive Databases Deductive databases are an extension of relational databases, in that in addition to manipulating explicitly represented facts, deductive databases provide ways to deduce facts from other facts in the database. We now give a very brief introduction to deductive databases. More details can be found in 8, 10, 17], Let L be a given underlying language with a nite set of constant, variable, and predicate symbols, but no function symbols. A term is either a variable or a constant. An atom is of the form p(t1 ; : : : ; tn), where p is a predicate symbol and the ti0 s are terms. A literal is either a positive literal A or a negative literal :A, where A is an atom. For a...
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