Fixed-Point Theory for Generalized Metric Spaces
111
4.6.14 Proposition Let l : B P → N be an arbitrary level mapping satisfying
the condition that l
−1 (n) is finite for each n ∈ N. If T P is non-expanding, then
T P is continuous in Q and hence is CS-continuous.
Proof: Let T P be non-expanding, and let (I n ) be a Cauchy sequence with
lim I n = I. Since T P is non-expansive, we obtain
0 ≤ d r (T P (I n ), T P (I)) ≤ d r (I n , I) → 0
and
0 ≤ d r (T P (I), T P (I n )) ≤ d r (I, I n ) → 0
by total boundedness of I P . By definition of d r and Proposition 4.6.11, it
follows that T P (I n ) is a Cauchy sequence and, by Proposition 3.3.5 and the
previous inequalities, T P (I n ) converges in Q to T P (I). Hence, lim T P (I n ) =
T P (I), again by Corollary 4.6.12.
•
We close with a brief discussion of several simple examples illustrating the
methods and results of this section as applied to normal logic programs P
relative to T P defined on I P . For full details the reader is again referred to
[Seda, 1997]. Thus, suppose that P is a normal logic program, that d r is the
quasimetric determined by a level mapping l defined on B P and satisfying the
property that l
−1 (n) is finite for all n, and that T P is CS-continuous relative
to d r or equivalently that T P is continuous in the topology Q.
4.6.15 Example Consider again the program P of Example 3.2.3
p(a) ←
p(s(X)) ← p(X)
and define l on B P by l(p(s
n (a))) = n. Then we see that d r (T P (I 1 ), T P (I 2 )) ≤
1 d r (I 1 , I 2 ) for all I 1 , I 2 ∈ I P . Therefore, T P is a contraction and is continuous
2
in Q and, hence, is CS-continuous. Thus, Theorem 4.6.3 applies and produces
a unique fixed point of T P . Of course, this fixed point coincides with the usual
one produced by considering powers T
n (∅) of ∅.
P
4.6.16 Example Consider the program P
p(s(X), a) ← p(s(X), a)
with the level mapping l defined on B P by l(p(s
n (a), s
m (a))) = n + m. Then
it is readily checked that T P is non-expansive (and therefore continuous in
Q), but not contractive, relative to the quasimetric d r determined by l, since
it is easy to find distinct I 1 and I 2 such that d r (T P (I 1 ), T P (I 2 )) = d r (I 1 , I 2 ).
Thus, Theorem 4.6.3 is applicable and, needless to say, produces numerous
fixed points of T P . For this reason, it follows that T P cannot be a contraction
relative to any metric. Thus, the approach to finding fixed points based on
metrics and the Banach contraction mapping theorem fails even for the rather
simple program P .
111
4.6.14 Proposition Let l : B P → N be an arbitrary level mapping satisfying
the condition that l
−1 (n) is finite for each n ∈ N. If T P is non-expanding, then
T P is continuous in Q and hence is CS-continuous.
Proof: Let T P be non-expanding, and let (I n ) be a Cauchy sequence with
lim I n = I. Since T P is non-expansive, we obtain
0 ≤ d r (T P (I n ), T P (I)) ≤ d r (I n , I) → 0
and
0 ≤ d r (T P (I), T P (I n )) ≤ d r (I, I n ) → 0
by total boundedness of I P . By definition of d r and Proposition 4.6.11, it
follows that T P (I n ) is a Cauchy sequence and, by Proposition 3.3.5 and the
previous inequalities, T P (I n ) converges in Q to T P (I). Hence, lim T P (I n ) =
T P (I), again by Corollary 4.6.12.
•
We close with a brief discussion of several simple examples illustrating the
methods and results of this section as applied to normal logic programs P
relative to T P defined on I P . For full details the reader is again referred to
[Seda, 1997]. Thus, suppose that P is a normal logic program, that d r is the
quasimetric determined by a level mapping l defined on B P and satisfying the
property that l
−1 (n) is finite for all n, and that T P is CS-continuous relative
to d r or equivalently that T P is continuous in the topology Q.
4.6.15 Example Consider again the program P of Example 3.2.3
p(a) ←
p(s(X)) ← p(X)
and define l on B P by l(p(s
n (a))) = n. Then we see that d r (T P (I 1 ), T P (I 2 )) ≤
1 d r (I 1 , I 2 ) for all I 1 , I 2 ∈ I P . Therefore, T P is a contraction and is continuous
2
in Q and, hence, is CS-continuous. Thus, Theorem 4.6.3 applies and produces
a unique fixed point of T P . Of course, this fixed point coincides with the usual
one produced by considering powers T
n (∅) of ∅.
P
4.6.16 Example Consider the program P
p(s(X), a) ← p(s(X), a)
with the level mapping l defined on B P by l(p(s
n (a), s
m (a))) = n + m. Then
it is readily checked that T P is non-expansive (and therefore continuous in
Q), but not contractive, relative to the quasimetric d r determined by l, since
it is easy to find distinct I 1 and I 2 such that d r (T P (I 1 ), T P (I 2 )) = d r (I 1 , I 2 ).
Thus, Theorem 4.6.3 is applicable and, needless to say, produces numerous
fixed points of T P . For this reason, it follows that T P cannot be a contraction
relative to any metric. Thus, the approach to finding fixed points based on
metrics and the Banach contraction mapping theorem fails even for the rather
simple program P .