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Unformatted text preview: n disgaree with each other, sometimes strongly.
Logics dealing with inconsistent information are called paraconsistent logics, and were
studied in detail by de Costa 7] and Belnap 2]. Blair and Subrahmanian 4] proposed
logic programming based on paraconsistent logic and Subrahmanian 22] extended the
work to disjunctive deductive databases.
In this paper, we present a generalisation of the relational data model. Our model
is based on the 4-valued paraconsistent logic of 2] and is capable of manipulating incomplete as well as inconsistent information. The incompleteness is at the tuple level,
in that whether or not a particular tuple belongs to a relation may not be known. This
notion of incompleteness is di erent from the other null-values related notions mentioned
earlier. Similarly, inconsistency is also at the tuple level, in that a particular tuple may
be considered to be both in and out of a relation.
We introduce paraconsistent relations, which are the fundamental mathematical structures underlying our model. A paraconsistent relation essentially contains two kinds of
2 tuples: ones that de nitely belong to the relation and others that de nitely do not belong
to the relation. These structures are strictly more general than ordinary relations, in that
for any ordinary relation there is a paraconsistent relation with the same information
content, but not vice versa. We de ne algebraic operators over paraconsistent relations
that extend the standard operators, such as selection, join, union, over ordinary relations.
In addition to answering queries in databases, we show another application of our
algebra on paraconsistent relations. We present a bottom-up method to construct the
weak well-founded model of general deductive databases; this model was proposed by
Fitting in 9].
The rest of the paper is organised as follows. Section 2 introduces paraconsistent
relations and two notions of generalising the usual relational operators, such as union,
join, projection, for these relations. Section 3 presents some actual generalised algebraic
operators for paraconsistent relations. These operators can be used for specifying queries
for database systems built on such relations. As another interesting application of these
operators, Section 4 gives a method for costructing the weak well-founded model of general
deductive databases. An important step in the construction is to translate the database
clauses into expressions involving the algebraic operators on paraconsistent relations.
Finally, Section 5 contains some concluding remarks and directions for future work. 2 Paraconsistent Relations
In this section, we construct a set-theoretic formulation of paraconsistent relations. Unlike
ordinary relations that can model worlds in which every tuple is known to either hold a
certain underlying predicate or to not hold it, paraconsistent relations provide a framework
for incomplete or even inconsistent information about tuples. They naturally model
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