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Unformatted text preview: onsistent partial interpretations de ned as follows:
13 De nition 19 Let I be a partial interpretation. Then TPF (I ) is a partial interpretation, given by F
TP (I )+ = fa j for some clause a l1; : : : ; lm in P ?, for each i; 1 i m,
if li is positive then li 2 I +, and
if li is negative then li0 2 I ?g;
TP (I )? = fa j for every clause a l1; : : : ; lm in P ?, there is some i; 1 i m,
such that if li is positive then li 2 I ?, and
if li is negative then li0 2 I +g: 2
It can be shown that TP preserves consistency and always possesses a least xpoint. This
least xpoint is called the weak well-founded model for P . The model can also be shown
to be TP " !, where the ordinal powers of TP are de ned as follows: De nition 20 For any ordinal ,
> h;; ;i
F " = T F (T F " ( ? 1))
: h < (T " )+;
P < if = 0,
if is a successor ordinal,
(TP " )?i if is a limit ordinal. 2 In 1], the upward closure ordinal of the immediate consequence function is de ned as
the least ordinal such that TP " is a xpoint of TP . The following observation for
deductive databases is relevant: Proposition 7 For any general deductive database P , the upward closure ordinal of TPF
is nite, i.e. there is a number n 0, such that TP " n = TP " !. 2
Thus a mechanism that \computes" the ordinal powers of TP can be employed to construct
the weak well-founded model of P .
Construction of the Weak Well-founded Model We now describe a method for constructing the weak well-founded model for a given
general deductive database P . In this model, paraconsistent relations are the semantic
objects associated with the predicate symbols occurring in P .
The method involves two steps. The rst step is to convert P into a set of paraconsistent relation de nitions for the predicate symbols occurring in P . These de nitions are
of the form
p = Dp ;
14 where p is a predicate symbol of P , and Dp is an algebraic expression involving predicate
symbols of P and paraconsistent relation operators. The second step is to iteratively
evaluate the expressions in these de nitions to incrementally construct the paraconsistent
relations associated with the predicate symbols.
A scheme is a Herbrand scheme if dom(A) is the Herbrand Universe, for all A 2 .
The schemes of the paraconsistent relations that we associate with the predicate symbols
are set internally. Let ? = h 1; 2 ; : : :i be an in nite sequence of some distinct attribute
names. For any n 1, let ?n be the Herbrand scheme f 1 ; : : : ; ng. We use the following
scheme renaming operators. De nition 21 Let = fA1 ; : : : ; Ang be any Herbrand scheme. Then,
(a) for any paraconsistent relation R on scheme ?n, R(A1 ; : : : ; An) is the paraconsistent
relation h ft 2 ( ) j for some t0 2 R+; t(Ai) = t0 ( i); for all i; 1 i ng;
ft 2 ( ) j for some t0 2 R?; t(Ai) = t0 ( i); for all i; 1 i ng i on scheme , and
(b) for any paraconsistent relation R on scheme , R A1 ; : : : ; An] is the...
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