paraconsistent

Consistent 2 5 n1 i 1 n p c t comps 2o

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Unformatted text preview: 1; : : : ; n, P( ) : comps 2O ( ) C( ) XXXX T( ) XXXXX XXXXX XXXX O( ) XX z Figure 1: Classes of relations De nition 7 A consistency preserving operator on paraconsistent relations with signature h 1 ; : : : ; n+1i is a strong generalisation of an operator on ordinary relations with the same signature, if for any consistent relations R1 ; : : : ; Rn on schemes 1 ; : : : ; n, respectively, we have comps +1 ( n (R1; : : : ; Rn)) = S ( )(comps 1 (R1 ); : : : ; comps (Rn)): 2 n Given an operator on ordinary relations, the behavior of a weak generalisation of is `controlled' only over the total relations. On the other hand, the behavior of a strong generalisation is `controlled' over all consistent relations. This itself suggests that strong generalisation is a stronger notion than weak generalisation. The following proposition makes this precise. Proposition 2 If is a strong generalisation of , then is also a weak generalisation of . Proof Let h 1 ; : : : ; n+1 i be the signature of and , and let R1 ; : : : ; Rn be any total relations on schemes 1 ; : : : ; n, respectively. Since all total relations are consistent, and is a strong generalisation of , we have that comps (R1; : : : ; Rn)) = S ( )(comps 1 (R1 ); : : : ; comps (Rn)): Proposition 1 gives us that for each i, 1 i n, comps (Ri ) is the singleton set fRi+g, i.e. f (Ri)g. Therefore, S ( )(comps 1 (R1 ); : : : ; comps (Rn )) is just the singleton set +1 ( n n i i n 6 (Rn))g. Hence, (R1 ; : : : ; Rn) is total, and +1 ( (R1; : : : ; Rn)) = ( 1 (R1 ); : : : ; (Rn)), i.e. is a weak generalisation of . 2 Though there may be many strong generalisations of an operator on ordinary relations, they all behave the same when restricted to consistent relations. In the next section, we propose strong generalisations for the usual operators on ordinary relations. The proposed generalised operators on paraconsistent relations correspond to the belief system intuition behind paraconsistent relations. f ( 1 (R1 ); : : : ; n n n 3 Algebraic Operators on Paraconsistent Relations In this section, we present one strong generalisation each for the usual ordinary relation operators, such as union, join, projection. To re ect generalisation, a dot is placed over an ordinary relation operator to obtain the corresponding paraconsistent relation operator. _ For example, 1 denotes the natural join among ordinary relations, and 1 denotes natural join on paraconsistent relations. Set-Theoretic Operators We rst introduce two fundamental set-theoretic algebraic operators on paraconsistent relations: De nition 8 Let R and S be paraconsistent relations on scheme . Then, (a) the union of R and S , denoted R _ S , is a paraconsistent relation on scheme , given by (R _ S )+ = R+ S +; (R _ S )? = R? \ S ?; _ (b) the complement of R, denoted ? R, is a paraconsistent relation on scheme , given by _ _ (? R ) + = R ? ; (? R ) ? = R + : 2 An intuitive appreciation of the union operator may be obtained by interpreting relations as proper...
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