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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...
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This note was uploaded on 09/28/2013 for the course CSC 8710 taught by Professor Staff during the Fall '08 term at Georgia State University, Atlanta.

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