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Logic Programming and Artificial Neural Networks
FIGURE 7.3: Sketch of a 3-layer recurrent network containing, from left to
right, 3 input, 4 hidden, and 3 output units and showing also the recurrent
connections from output layer to input layer.
also that a recurrent network can perform iterated computations because the
output values can be returned to the input layer via the connections just described; it can thus perform computation of the iterates T
k (I), k ∈ N, for
example, where I is an interpretation and T is a semantic operator.
7.3 The Core Method as a General Approach to
Integration
In this section, we outline the idea underlying the approach presented below. Suppose given a normal logic program P and any one of the semantic
operators T P : I P → I P we have thus far associated with P , using T P and
I P as generic symbols for a semantic operator and its underlying set of interpretations. For simplicity, we assume the interpretations in question are
Herbrand interpretations taking values in a truth set T , although the conclusions we make here are valid over any preinterpretation J whose domain
D is countable. Can one find, or at least show the existence of, a multilayer
feedforward network F P which computes T P in some sense? Furthermore, can
this network F P , or some other appropriate network, compute the least fixed
point of T P assuming the least fixed point of T P exists?
A few general remarks are in order at this point. To begin with, multilayer feedforward networks, even 3-layer feedforward networks, are known to
be extremely powerful computing devices and indeed are known to be universal approximators in the sense made precise in the statement of Funahashi’s
theorem, Theorem 7.2.2, earlier.
11 Therefore, one might expect them to have
the ability to carry out the required computations, and this is so. Indeed, suppose that P is a first-order program and endow I P with the Cantor topology,
11 See [Funahashi, 1989, Hornik et al., 1989] for full details.
Logic Programming and Artificial Neural Networks
FIGURE 7.3: Sketch of a 3-layer recurrent network containing, from left to
right, 3 input, 4 hidden, and 3 output units and showing also the recurrent
connections from output layer to input layer.
also that a recurrent network can perform iterated computations because the
output values can be returned to the input layer via the connections just described; it can thus perform computation of the iterates T
k (I), k ∈ N, for
example, where I is an interpretation and T is a semantic operator.
7.3 The Core Method as a General Approach to
Integration
In this section, we outline the idea underlying the approach presented below. Suppose given a normal logic program P and any one of the semantic
operators T P : I P → I P we have thus far associated with P , using T P and
I P as generic symbols for a semantic operator and its underlying set of interpretations. For simplicity, we assume the interpretations in question are
Herbrand interpretations taking values in a truth set T , although the conclusions we make here are valid over any preinterpretation J whose domain
D is countable. Can one find, or at least show the existence of, a multilayer
feedforward network F P which computes T P in some sense? Furthermore, can
this network F P , or some other appropriate network, compute the least fixed
point of T P assuming the least fixed point of T P exists?
A few general remarks are in order at this point. To begin with, multilayer feedforward networks, even 3-layer feedforward networks, are known to
be extremely powerful computing devices and indeed are known to be universal approximators in the sense made precise in the statement of Funahashi’s
theorem, Theorem 7.2.2, earlier.
11 Therefore, one might expect them to have
the ability to carry out the required computations, and this is so. Indeed, suppose that P is a first-order program and endow I P with the Cantor topology,
11 See [Funahashi, 1989, Hornik et al., 1989] for full details.