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content, but not vice versa. Since paraconsistent relations are capable of containing
contradictory information, these structures model belief systems more naturally than
knowledge systems.
We developed two notions of generalising operators on ordinary relations for paraconsistent relations. Of these, the stronger notion guarantees that any generalised operator is
\well-behaved" for paraconsistent relation operands that contain consistent information.
For some well-known operators on ordinary relations, such as union, join, projection,
we introduced generalised operators on paraconsistent relations. These generalised operators maintain the belief system intuition behind paraconsistent relations, and are shown
to be \well-behaved" in the sense mentioned above.
Our data model can be used to represent relational information that may be incomplete
or inconsistent. As usual, the algebraic operators can be used to construct queries to any
database systems for retrieving paraconsistent information. As another application of
paraconsistent relations and the algebra on them, we presented a method for constructing
the weak well-founded model for general deductive databases 9]. This method requires
translating the clauses of the database into expressions involving the generalised operators
introduced earlier. A minor modi cation to this method can tailor it for constructing the
well-founded model for such databases 25].
Recently there has been some interest in studying extended logic programs in which
the head of clauses can have one or more literals 19]. This leads to two notions of negation:
implicit negation (corresponding to negative literals in the body) and explicit negation
(corresponding to negative literals in the head). One possible direction for further research is to extend the paraconsistent relational model presented in this paper to include
disjunctive tuples as in 16], thereby providing a framework under which the semantics of
extended logic programs could be constructed in a bottom-up manner. Allowing explicit
negation in a disjunctive deductive database/logic program usually cuts down on the
number of minimal models, sometimes quite drastically, and as a consequence increases
the e ciency of query processing. This will be one of the main motivations in exploring
the possibility of paraconsistent relations with disjunctive tuples. 18 References
1] R. Bagai, M. Bezem, and M. H. van Emden. On downward closure ordinals of logic
programs. Fundamenta Informaticae, XIII(1):67{83, 1990.
2] N. D. Belnap. A useful four-valued logic. In G. Eppstein and J. M. Dunn, editors,
Modern Uses of Many-valued Logic, pages 8{37. Reidel, Dordrecht, 1977.
3] J. Biskup. A foundation of Codd's relational maybe-operations. ACM Transactions
on Database Systems, 8(4):608{636, December 1983.
4] H. A. Blair and V. S. Subrahmanian. Paraconsistent logic programming. Theoretical
Computer Science, 68:135{154, 1989.
5] E. F. Codd. A relational mo...

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