219
Logic Programming and Artificial Neural Networks
Theorem 7.7.1 covers, among others, all Fitting-style operators from Section 5.2.1.
The fixpoint completion, studied in Section 6.1, also turns out to be very
useful, since it allows us to reduce treatments of the Gelfond–Lifschitz operator to the single-step operator. According to Theorem 7.2.2, we are first of
all interested in carrying over continuity results with respect to the Cantor
topology. From Theorem 6.2.2 we thus obtain the following result.
7.7.2 Theorem Let P be a normal logic program, and let the following condition be satisfied for all I ∈ I P and A ∈ B P : whenever GL P (I)(A) = f ,
then either there is no clause with head A in ground(P ) or there exists a
finite set S(I, A) = {A 1 , . . . , A k } ⊆ B P such that I(A i ) = t for all i, and
for every clause A ← body in ground(P ) at least one ¬A i or some B with
GL P (I)(B) = f occurs in body. Then GL P can be uniformly approximated
by 3-layer feedforward networks in the sense of Theorem 7.2.2.
We also obtain the following corollary, taking Corollary 6.2.3 into account.
7.7.3 Corollary Let P be a covered normal logic program. Then GL P can
be uniformly approximated by 3-layer feedforward networks in the sense of
Theorem 7.2.2.
Likewise, the remark given in Footnote 3 on page 170 of Chapter 6 together
with Lemma 5.4.12 allows us to derive similar characterizations of continuity
for the operator characterizing three-valued stable models.
In principle, one can use the results recorded earlier to embark on investigations similar to those undertaken in Sections 7.5.3 to 7.5.5, for example.
However, a direct application of the results for the Gelfond–Lifschitz operator is hardly satisfactory since the computation of the fixpoint completion
can only be carried out in an approximate manner. How one deals with this
problem, and what it entails, remains to be investigated.
It should be clear from Section 7.6 how one carries over the approach
using a finite subset of the grounding of a program to other locally finite
local consequence operators. For operators like the Gelfond–Lifschitz operator,
however, a straightforward approach is rather unsatisfactory due to the fact
that one iteration of the Gelfond–Lifschitz operator involves the taking of
a limit of the single-step operator for definite programs. As an alternative
approach, we could again first compute the fixpoint completion of the program
and employ Theorem 6.1.4 in conjunction with the methods from Section 7.6,
but alas, we have noted already that computation of the fixpoint completion
can only be done in an approximate manner. How to deal with this problem in
an appropriate manner again is something which remains to be investigated.
Before closing this chapter, we would like to remark that there is a plethora
of work which has been done on the integration of logic and neural networks.
43
43 See, for example, [Bader and Hitzler, 2005, Hammer and Hitzler, 2007] for overviews.
Logic Programming and Artificial Neural Networks
Theorem 7.7.1 covers, among others, all Fitting-style operators from Section 5.2.1.
The fixpoint completion, studied in Section 6.1, also turns out to be very
useful, since it allows us to reduce treatments of the Gelfond–Lifschitz operator to the single-step operator. According to Theorem 7.2.2, we are first of
all interested in carrying over continuity results with respect to the Cantor
topology. From Theorem 6.2.2 we thus obtain the following result.
7.7.2 Theorem Let P be a normal logic program, and let the following condition be satisfied for all I ∈ I P and A ∈ B P : whenever GL P (I)(A) = f ,
then either there is no clause with head A in ground(P ) or there exists a
finite set S(I, A) = {A 1 , . . . , A k } ⊆ B P such that I(A i ) = t for all i, and
for every clause A ← body in ground(P ) at least one ¬A i or some B with
GL P (I)(B) = f occurs in body. Then GL P can be uniformly approximated
by 3-layer feedforward networks in the sense of Theorem 7.2.2.
We also obtain the following corollary, taking Corollary 6.2.3 into account.
7.7.3 Corollary Let P be a covered normal logic program. Then GL P can
be uniformly approximated by 3-layer feedforward networks in the sense of
Theorem 7.2.2.
Likewise, the remark given in Footnote 3 on page 170 of Chapter 6 together
with Lemma 5.4.12 allows us to derive similar characterizations of continuity
for the operator characterizing three-valued stable models.
In principle, one can use the results recorded earlier to embark on investigations similar to those undertaken in Sections 7.5.3 to 7.5.5, for example.
However, a direct application of the results for the Gelfond–Lifschitz operator is hardly satisfactory since the computation of the fixpoint completion
can only be carried out in an approximate manner. How one deals with this
problem, and what it entails, remains to be investigated.
It should be clear from Section 7.6 how one carries over the approach
using a finite subset of the grounding of a program to other locally finite
local consequence operators. For operators like the Gelfond–Lifschitz operator,
however, a straightforward approach is rather unsatisfactory due to the fact
that one iteration of the Gelfond–Lifschitz operator involves the taking of
a limit of the single-step operator for definite programs. As an alternative
approach, we could again first compute the fixpoint completion of the program
and employ Theorem 6.1.4 in conjunction with the methods from Section 7.6,
but alas, we have noted already that computation of the fixpoint completion
can only be done in an approximate manner. How to deal with this problem in
an appropriate manner again is something which remains to be investigated.
Before closing this chapter, we would like to remark that there is a plethora
of work which has been done on the integration of logic and neural networks.
43
43 See, for example, [Bader and Hitzler, 2005, Hammer and Hitzler, 2007] for overviews.