153
Supported Model Semantics
TABLE 5.1: Several
p q
u u
truth tables for
p ∧ 1 q p ∧ 2 q
u
u
three-valued
p ∧ 3 q
u
logics.
u f
f
u
u
u t
f u
u
f
u
f
u
u
f f
f
f
f
f t
t u
f
u
f
u
f
u
t f
f
f
f
t t
t
t
t
p q p ∨ 1 q p ∨ 2 q
u u
u
u
u f
u
u
u t
t
u
f u
u
u
f f
f
f
f t
t
t
t u
t
u
t f
t
t
t t
t
t
p ¬p
u u
f
t
t f
all of the body i are false, is undefined if and only if one of the
body i is undefined, and is true otherwise.
Finally, if A is an atom which does not appear as the head of a clause in
ground(P ), then we say, by abuse of notation, that
o A
body is the
←
pseudo-clause associated with A, and we take i body
∈∅
i
with respect to 1 and with respect to 2 . Notice now that
o
i∈∅
i
to be false both
∨
∨
every element
A of B P is the head of the pseudo-clause associated with A and that this
pseudo-clause is uniquely determined by P for a given A.
The following notation will be convenient. Let A ∈ B P , let body A be the
body of the pseudo-clause associated with A, and let I be a three-valued interpretation. Then write I j,k (body A ) for the truth value, under I, of body A with
respect to ∧ j and ∨ k , for j = 1, 2, 3 and k = 1, 2. The following proposition
follows easily from the definitions.