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**Unformatted text preview: **ny literal l we let l0 denote its complementary literal, i.e. if l is positive then l0 = :l,
otherwise l = :l0. De nition 13 A deductive database is a nite set of clauses of the form
a b 1 ; b2 ; : : : ; b m
where m 0 and a and each bi is an atom. 2
A term, atom, literal, or clause is called ground if it contains no variables. The
Herbrand Universe of the underlying language is the set of all ground terms; the Herbrand
Base of the language is the set of all ground atoms; A Herbrand Interpretation of the
language is any subset of the Herbrand Base. A ground instance of a term, atom, literal,
or clause Q is the term, atom, literal, or clause, respectively, obtained by replacing each
variable in Q by a constant. For any deductive database P , we let P ? denote the set
of all ground instances of clauses in P . Note that since the underlying language has no
function symbols, unlike logic programs, P ? is always nite.
One way to give the semantics of a de nite deductive database is the least xpoint of
the following immediate consequence function TP on Herbrand interpretations: De nition 14 Let I be a Herbrand interpretation for a deductive database P . Then
TP (I ) is a Herbrand interpretation, given by
TP (I ) = fa j for some clause a b1 ; : : : ; bm in P ?; fb1 ; : : : ; bmg I g: 2
12 It is well-known that TP always possesses a least xpoint with respect to the partial order
of set inclusion. The least xpoint can be shown to be the minimal model for P . This
model is also known to be TP " !, where the ordinal powers of TP are de ned as follows: De nition 15 For any ordinal ,
8
>;
if = 0,
<
TP " = > TP (TP " ( ? 1)) if is a successor ordinal,
: < (TP " );
if is a limit ordinal. 2
The expressive power of the clauses of a deductive databases can be increased by
allowing negated atoms in their bodies. This results in a more general class of deductive
databases de ned below. De nition 16 A general deductive database is a nite set of clauses of the form
a l 1 ; l2 ; : : : ; l m where a is an atom, m 0 and each li is a literal. 2
Weak Well-founded Model of General Deductive Databases One of the semantics given for general deductive databases is described next. It was
originally presented in 9] for logic programs. De nition 17 A partial interpretation is a pair I = hI +; I ?i, where I + and I ? are any
subsets of the Herbrand Base. 2 A partial interpretation I is consistent if I + \ I ? = ;. For any partial interpretations
I and J , we let I \ J be the partial interpretation hI + \ J +; I ? \ J ?i, and I J be the
partial interpretation hI + J +; I ? J ?i. We also say I J whenever I + J + and
I ? J ?. De nition 18 Let S be a set partially ordered by . A map T : S ! S is monotonic if,
for any X; Y 2 S , X Y implies T (X ) T (Y ). 2
For any general deductive database P , recall that P ? is the set of all ground instances
of clauses in P . The weak well-founded model of P is the least xpoint of the immediate
F
consequence function TP on c...

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