209
Logic Programming and Artificial Neural Networks
Just as for the network architectures described previously, we can train
vector-based networks using a set of input-output pairs. The position of the
units, that is, the weights between the input and hidden layer, are modified
such that a unit is located in the centre of all the inputs it is responsible for.
The output weights are trained such that they represent the average output
of all inputs within the unit’s responsibility. If, furthermore, two neighbouring
units have similar output weights, then one of them can be removed because
the other unit will take over in that eventuality. A unit whose accumulated
error is very large can be replaced by two units that can be adapted independently, thus allowing the network to refine its input-output function in certain
areas.
The first experiments which reported on this approach
36 showed the applicability of this learning method in the area of neural-symbolic integration.
A randomly initialized network was trained using the embedded versions of
an interpretation I as input values and of T P (I) as output values for a given
program P . The network learned the mapping and could be used iteratively
by adding recurrent connections between the output and input layers.
7.5.6 Approximating the (Least) Fixed Point of T P
Thus far, we have discussed at some length the issue of the approximate
computation of the T P -operator for first-order normal logic programs P . We
turn now to discussing, fairly briefly, the question of the approximate computation of its fixed points. One approach is to carry forward the work of the
previous sections and employ iterates of (recurrent) neural networks which
approximate T P to approximate iterates of the operator T P , but, as already
noted earlier, the problem then emerges of uniformly controlling the error
estimates under iteration.
37
On the other hand, one can approach the problem of computing the least
fixed point of T P for arbitrary definite logic programs P by a modification of
the previous approach employing the subset P n of ground(P ), except that we
do not assume that P is covered, and instead we ensure that the appropriate
subset of ground(P ) is finite by other means.
Thus, let P denote an arbitrary (first-order) definite logic program, and
denote by I the least fixed point of T P . Let l : B P → ω be a level mapping
with the property that l
−1 (n) is a finite set for each n ∈ ω. We proceed to
sketch the details of the construction of a finite subset P n of ground(P ), where
n is a given natural number, which will play the sort of role here that P n plays
in Proposition 7.5.5 and its companion results.
38 We start with the following
claim.
36 See [Bader et al., 2007].
37 This point is discussed in [Hitzler et al., 2004, Section 4.3], but quite strong conditions,
for example, Lipschitz continuity [Hitzler et al., 2004, Theorem 4.19], are required for things
to work satisfactorily.
38 See [Seda, 2006] for full details.
Logic Programming and Artificial Neural Networks
Just as for the network architectures described previously, we can train
vector-based networks using a set of input-output pairs. The position of the
units, that is, the weights between the input and hidden layer, are modified
such that a unit is located in the centre of all the inputs it is responsible for.
The output weights are trained such that they represent the average output
of all inputs within the unit’s responsibility. If, furthermore, two neighbouring
units have similar output weights, then one of them can be removed because
the other unit will take over in that eventuality. A unit whose accumulated
error is very large can be replaced by two units that can be adapted independently, thus allowing the network to refine its input-output function in certain
areas.
The first experiments which reported on this approach
36 showed the applicability of this learning method in the area of neural-symbolic integration.
A randomly initialized network was trained using the embedded versions of
an interpretation I as input values and of T P (I) as output values for a given
program P . The network learned the mapping and could be used iteratively
by adding recurrent connections between the output and input layers.
7.5.6 Approximating the (Least) Fixed Point of T P
Thus far, we have discussed at some length the issue of the approximate
computation of the T P -operator for first-order normal logic programs P . We
turn now to discussing, fairly briefly, the question of the approximate computation of its fixed points. One approach is to carry forward the work of the
previous sections and employ iterates of (recurrent) neural networks which
approximate T P to approximate iterates of the operator T P , but, as already
noted earlier, the problem then emerges of uniformly controlling the error
estimates under iteration.
37
On the other hand, one can approach the problem of computing the least
fixed point of T P for arbitrary definite logic programs P by a modification of
the previous approach employing the subset P n of ground(P ), except that we
do not assume that P is covered, and instead we ensure that the appropriate
subset of ground(P ) is finite by other means.
Thus, let P denote an arbitrary (first-order) definite logic program, and
denote by I the least fixed point of T P . Let l : B P → ω be a level mapping
with the property that l
−1 (n) is a finite set for each n ∈ ω. We proceed to
sketch the details of the construction of a finite subset P n of ground(P ), where
n is a given natural number, which will play the sort of role here that P n plays
in Proposition 7.5.5 and its companion results.
38 We start with the following
claim.
36 See [Bader et al., 2007].
37 This point is discussed in [Hitzler et al., 2004, Section 4.3], but quite strong conditions,
for example, Lipschitz continuity [Hitzler et al., 2004, Theorem 4.19], are required for things
to work satisfactorily.
38 See [Seda, 2006] for full details.