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**Unformatted text preview: **paraconsistent
relation
h ft 2 (?n) j for some t0 2 R+; t( i) = t0 (Ai); for all i; 1 i ng;
ft 2 (?n) j for some t0 2 R?; t( i ) = t0(Ai ); for all i; 1 i ng i
on scheme ?n. 2
Before describing our method to convert the given database P into a set of de nitions
for the predicate symbols in P , let us look at an example. Suppose the following are the
only clauses with the predicate symbol p in their heads:
p(X)
p(Y) r(X,Y),
s(Y,a) :p(Y) From these clauses the algebraic de nition constructed for the symbol p is the following:
__
p = ( _ {X} (r(X; Y) 1 ?p(Y))) X] _ (s(Y; Z)))) Y]
Such a conversion exploits the close connection between attribute names in relation
schemes and variables in clauses, as pointed out in 24]. The expression thus constructed
can be used to arrive at a better approximation of the paraconsistent relation p from
some approximations of p, r and s.
The algebraic expression for the predicate symbol p is a union ( _ ) of the expressions
obtained from each clause containing the symbol p in its head. It therefore su ces to
give an algorithm for converting one clause into an expression.
15 Algorithm CONVERT
Input: A general deductive database clause l0 l1 ; : : : ; l m .
Let l0 be of the form p0(A01 ; : : : ; A0k0 ), and each li, 1 i m, be either of the form
pi(Ai1 ; : : : ; Aik ), or of the form :pi (Ai1; : : : ; Aik ). For any i, 0 i m, let Vi be the set
of all variables occurring in li .
Output: An algebraic expression involving paraconsistent relations.
Method: The expression is constructed by the following steps:
i i 1. For each argument Aij of literal li , construct argument Bij and condition Cij as
follows:
(a) If Aij is a constant a, then Bij is any brand new variable and Cij is Bij = a.
(b) If Aij is a variable, such that for each k, 1 k < j , Aik 6= Aij , then Bij is Aij
and Cij is true.
(c) If Aij is a variable, such that for some k, 1 k < j , Aik = Aij , then Bij is a
brand new variable and Cij is Aij = Bij .
2. Let ^i be the atom pi(Bi1 ; : : : ; Bik ), and Fi be the conjunction Ci1 ^ ^ Cik . If li
l
is a positive literal, then let Qi be the expression _ V ( _ F (^i)). Otherwise, let Qi be
l
_
the expression ? _ V ( _ F (^i)).
l
As a syntactic optimisation, if all conjuncts of Fi are true (i.e. all arguments of li
are distinct variables), then both _ F and _ V are reduced to identity operations,
and are hence dropped from the expression. For example, if li = :p(X,Y), then
_
Qi = ?p(X,Y).
_
3. Let E be the natural join (1) of the Qi 's thus obtained, 1 i m. The output
expression is ( _ F0 ( _ V (E ))) B01 ; : : : ; B0k0 ], where V is the set of variables occurring
in ^0.
l
As in step 2, if all conjuncts in F0 are true, then _ F0 is dropped from the output
expression. However, _ V is never dropped, as the clause body may contain variables
not in V . 2
i i i i i i i i From the algebraic expressions obtained by Algorithm CONVERT for clauses in the
given general deductive database, we construct a system of equations de ning par...

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