This preview shows page 1. Sign up to view the full content.
Unformatted text preview: rather than knowledge systems, and are thus a generalisation of ordinary
relations. The operators on ordinary relations can also be generalised for paraconsistent
relations. However, any such generalisation of operators should maintain the belief system
intuition behind paraconsistent relations. This section also develops two di erent notions
of operator generalisations.
Let a relation scheme (or just scheme) be a nite set of attribute names, where for
any attribute name A 2 , dom(A) is a non-empty domain of values for A. A tuple on
is any map t : ! A2 dom(A), such that t(A) 2 dom(A), for each A 2 . Let ( )
denote the set of all tuples on . De nition 1 An ordinary relation on scheme is any subset of ( ). We let O( ) be
the set of all ordinary relations on . 2 The above is the usual de nition of relations. We call them `ordinary' relations to distinguish them from other kinds of relations introduced below.
3 De nition 2 A paraconsistent relation on scheme is a pair R = hR+; R?i, where R+
and R? are any subsets of ( ). We let P ( ) be the set of all paraconsistent relations
on . 2 Intuitively, R+ may be considered as the set of all tuples for which R is believed to
be true, and R? the set of all tuples for which R is believed to be false. Note that
since contradictory beliefs are possible, we do not assume R+ and R? to be mutually
disjoint, though this condition holds in an important class of paraconsistent relations. As
paraconsistent relations may contain contradictory information, they model belief systems
more naturally than knowledge systems. Furthermore, R+ and R? may not together cover
all tuples in ( ). De nition 3 A paraconsistent relation R on scheme is called a consistent relation if
R+ \ R? = ;. We let C ( ) be the set of all consistent relations on . Moreover, R is
called a complete relation if R+ R? = ( ). If R is both consistent and complete, i.e.
R? = ( ) ? R+ , then it is a total relation, and we let T ( ) be the set of all total relations
on . 2
It should be observed that (the positive parts of) total relations are essentially the ordinary
relations. We make this relationship explicit by de ning a one-one correspondence :
T ( ) ! O( ), given by (hR+; R?i) = R+ . This correspondence is used frequently in
the following discussion.
Operator Generalisations It is easily seen that paraconsistent relations are a generalisation of ordinary relations, in
that for each ordinary relation there is a paraconsistent relation with the same information
content, but not vice versa. It is thus natural to think of generalising the operations
on ordinary relations, such as union, join, projection etc., to paraconsistent relations.
However, any such generalisation should be intuitive with respect to the belief system
model of paraconsistent relations. We now construct a framework for operators on both
kinds of relations and introduce two di erent notions of the generalisation relationship
among their operators.
An n-ary operator on ordinary relations wi...
View Full Document