83
Topology and Logic Programming
context within which it naturally arises; in particular, we provide necessary
and sufficient conditions for the continuity of T P in Q to hold. Some results are
also known which ensure discontinuity of T P , see [Seda, 1995], for example,
and we pause briefly to consider an interesting example of this.
3.3.11 Example Consider the program P consisting of the single clause
p ← ¬q(X), whose underlying first-order language L is assumed to contain a
constant symbol o, a function symbol s, and predicate symbols r and t in addition to the symbols present in P . For each binary sequence a = (a n ) n∈N (of 0s
and 1s), we form the set A a = {A 1 , A 2 , A 3 , ...}, where A i = r(s
i (o)) if a i = 0
and A i = t(s
i (o)) if a i = 1. Finally, let K n = {q(0), q(s(0)), ..., q(s
n (0))} for
each n ∈ N.
Then for each binary sequence a, the sequence of interpretations I n =
A a ∪ K n converges in Q to the interpretation I a = A a ∪ {q(s
n (o)) | n ∈ N}
by Theorem 3.3.4. On the other hand, T P (I n ) = {p}, whereas T P (I a ) = ∅.
Hence, T P (I n ) does not converge to T P (I a ) in Q, and so T P is discontinuous
at I a .
Since we have uncountably many binary sequences a, T P has uncountably
many points of discontinuity in Q.
3.4 Operators on Spaces of Valuations Revisited
Finally, we want to briefly return to the operators defined on I(X, T ),
which were discussed in Section 1.3.4, namely, the operators ¬, ∨, and ∧.
We have already noted in Section 1.3.4 that ¬ is not order continuous and,
hence, not Scott continuous relative to the orderings ≤ t , in which f ≤ t t. It
is, however, Scott continuous in the orderings ≤ k , as we now see. Of course,
one can similarly deal with other connectives such as → and ↔ in the same
way. However, as we have seen earlier, these are usually made to depend on
the three connectives we have already considered and therefore need not be
pursued further.
Our objective here is to examine the continuity of the operators ¬, ∨, and
∧ relative to the Scott and Cantor topologies, and we first deal with the Scott
topology. Again, we concentrate on the truth set FOU R for precisely the same
reasons as stated in Section 1.3.4.
3.4.1 Theorem Let T denote Belnap’s logic FOU R. Then the following
statements hold.
(a) The negation operator ¬ : I(X, T ) → I(X, T ) is continuous in the Scott
topology relative to the knowledge ordering [ k , but not relative to the
truth ordering [ t . The same statement is true in the case of Kleene’s
strong three-valued logic.
Topology and Logic Programming
context within which it naturally arises; in particular, we provide necessary
and sufficient conditions for the continuity of T P in Q to hold. Some results are
also known which ensure discontinuity of T P , see [Seda, 1995], for example,
and we pause briefly to consider an interesting example of this.
3.3.11 Example Consider the program P consisting of the single clause
p ← ¬q(X), whose underlying first-order language L is assumed to contain a
constant symbol o, a function symbol s, and predicate symbols r and t in addition to the symbols present in P . For each binary sequence a = (a n ) n∈N (of 0s
and 1s), we form the set A a = {A 1 , A 2 , A 3 , ...}, where A i = r(s
i (o)) if a i = 0
and A i = t(s
i (o)) if a i = 1. Finally, let K n = {q(0), q(s(0)), ..., q(s
n (0))} for
each n ∈ N.
Then for each binary sequence a, the sequence of interpretations I n =
A a ∪ K n converges in Q to the interpretation I a = A a ∪ {q(s
n (o)) | n ∈ N}
by Theorem 3.3.4. On the other hand, T P (I n ) = {p}, whereas T P (I a ) = ∅.
Hence, T P (I n ) does not converge to T P (I a ) in Q, and so T P is discontinuous
at I a .
Since we have uncountably many binary sequences a, T P has uncountably
many points of discontinuity in Q.
3.4 Operators on Spaces of Valuations Revisited
Finally, we want to briefly return to the operators defined on I(X, T ),
which were discussed in Section 1.3.4, namely, the operators ¬, ∨, and ∧.
We have already noted in Section 1.3.4 that ¬ is not order continuous and,
hence, not Scott continuous relative to the orderings ≤ t , in which f ≤ t t. It
is, however, Scott continuous in the orderings ≤ k , as we now see. Of course,
one can similarly deal with other connectives such as → and ↔ in the same
way. However, as we have seen earlier, these are usually made to depend on
the three connectives we have already considered and therefore need not be
pursued further.
Our objective here is to examine the continuity of the operators ¬, ∨, and
∧ relative to the Scott and Cantor topologies, and we first deal with the Scott
topology. Again, we concentrate on the truth set FOU R for precisely the same
reasons as stated in Section 1.3.4.
3.4.1 Theorem Let T denote Belnap’s logic FOU R. Then the following
statements hold.
(a) The negation operator ¬ : I(X, T ) → I(X, T ) is continuous in the Scott
topology relative to the knowledge ordering [ k , but not relative to the
truth ordering [ t . The same statement is true in the case of Kleene’s
strong three-valued logic.