**Unformatted text preview: **aconsistent relations as follows. De nition 22 For any general deductive database P , EQN(P ) is a set of all equations of the form p = Dp, where p is a predicate symbol of P , and Dp is the union ( _ ) of all
expressions obtained by Algorithm CONVERT for clauses in P with symbol p in their
head. The algebraic expression Dp is also called a de nition of p. 2
16 It is evident that a predicate symbol may have many de nitions. It can be shown that
the above method for converting a general deductive database P into de nitions for
F
its predicate symbols terminates, and that the de nitions produced mimic the TP map
de ned in 9].
The second and nal step in our model construction process is to incrementally construct the paraconsistent relations de ned by the given database. For any general deductive database P , we let PE and PI denote its extensional and intensional portions,
respectively. PE is essentially the set of clauses of P with empty bodies, and PI is the set
of all other clauses of P . Without loss of generality, we assume that no predicate symbol
?
occurs both in PE as well as in PI . Let us recall that PE is the set of all ground instances
of clauses in PE .
The overall construction algorithm is rather straightforward. It treats the predicate
symbols in a given database as imperative \variable names" that may contain a paraconsistent relation as value. Thus, any variable p has two set-valued elds, namely p+ and
p? . Algorithm CONSTRUCT
Input: A general deductive database P .
Output: Paraconsistent relation values for the predicate symbols of P .
Method: The values are computed by the following steps:
1. (Initialisation)
(a) Compute EQN(PI ) using Algorithm CONVERT for each clause in PI .
(b) For each predicate symbol p in PE , set
p+
p? ?
= fha1; : : : ; ak i j p(a1 ; : : : ; ak )
2 PE g; and
= fhb1 ; : : : ; bk i j k is the arity of p, and p(b1 ; : : : ; bk ) ?
62 PE g: (c) For each predicate symbol p in PI , set p+ = ;, and p? = ;.
2. For each equation of the form p = Dp in EQN(PI ), compute the expression Dp and
set p to the resulting paraconsistent relation.
3. If step 2 involved a change in the value of some p, goto 2.
4. Output the nal values of all predicate symbols in PE and PI . 2
Again, we omit the proof of termination of Algorithm CONSTRUCT and that it
constructs the weak well-founded model of the given database.
17 5 Conclusions and Future Work
We have presented a generalisation of the relational data model, which is capable of
manipulating incomplete or even inconsistent information. Paraconsistent relations, based
on Belnap's 4-valued logic 2], form the mathematical structures underlying the model.
A paraconsistent relation essentially contains two kinds of tuples: ones for which an
underlying predicate is believed to be true, and ones for which that predicate is believed
to be false. These structures are strictly more general than ordinary relations, in that
for any ordinary relation there is a paraconsistent relation wi...

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