This preview shows page 1. Sign up to view the full content.
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
View Full Document