196
Mathematical Aspects of Logic Programming Semantics
where 1 ≤ i ≤ n and i is even. The least model for P 4 maps each atom to t
and is computed in five steps by the recurrent network corresponding to P 4 .
We note that the networks constructed by the translation algorithm presented previously cannot be trained by the usual learning methods applied
to connectionist systems. It was observed in [d’Avila Garcez et al., 1997] (see
also [d’Avila Garcez and Zaverucha, 1999, d’Avila Garcez et al., 2002]) that
results similar to Theorem 7.4.1 and Corollary 7.4.3 can be obtained if the
binary threshold units occurring in the hidden layer of the feedforward kernels
are replaced by sigmoidal units. We omit the technical details here and refer
to the above cited literature. Such a move renders the kernels accessible to
the backpropagation algorithm, a standard technique for training feedforward
networks [Rumelhart et al., 1986].
7.5 First-Order Programs
A central problem for neural-symbolic integration is the determination of
a good representation of first-order rules within a connectionist setting. Such
a representation would result, at least, in the computation or approximation
of the associated semantic operators. That approximating networks exist for
the immediate consequence operators of acyclic logic programs was the first
result obtained in this regard, see [H¨ olldobler et al., 1999], but it was shown
with the help of Funahashi’s theorem, which is non-constructive as we have
already observed. In this section, we outline the ideas underlying the general
problem and also discuss different constructive approaches to it. But before
going into details, we need to answer the following questions.
• Why do we need to approximate operators such as the T P -operator?
• What does approximation mean in our context?
The first question is easily answered: even a single application of the T P
operator can lead to infinite results. For example, assume P is a program
containing the fact p(X). Applying the T P -operator once (to an arbitrary
interpretation) leads to a result containing infinitely many atoms, namely, all
p(X)-atoms for every X. In this simple example, we might be able to represent
this particular result in a finite way, but things might become arbitrarily
complex for other programs using the same or similar representations.
21
21 Indeed, the so-called rational models were developed to tackle this representational
problem for certain programs, see [Bornscheuer, 1996]. Unfortunately, there is no way to
compute an upper bound on the size of this rational representation, and hence it does not
give us any immediate advantages. Because we are not aware of any other finite representation, we will concentrate here on the standard representation using Herbrand interpretations.
Mathematical Aspects of Logic Programming Semantics
where 1 ≤ i ≤ n and i is even. The least model for P 4 maps each atom to t
and is computed in five steps by the recurrent network corresponding to P 4 .
We note that the networks constructed by the translation algorithm presented previously cannot be trained by the usual learning methods applied
to connectionist systems. It was observed in [d’Avila Garcez et al., 1997] (see
also [d’Avila Garcez and Zaverucha, 1999, d’Avila Garcez et al., 2002]) that
results similar to Theorem 7.4.1 and Corollary 7.4.3 can be obtained if the
binary threshold units occurring in the hidden layer of the feedforward kernels
are replaced by sigmoidal units. We omit the technical details here and refer
to the above cited literature. Such a move renders the kernels accessible to
the backpropagation algorithm, a standard technique for training feedforward
networks [Rumelhart et al., 1986].
7.5 First-Order Programs
A central problem for neural-symbolic integration is the determination of
a good representation of first-order rules within a connectionist setting. Such
a representation would result, at least, in the computation or approximation
of the associated semantic operators. That approximating networks exist for
the immediate consequence operators of acyclic logic programs was the first
result obtained in this regard, see [H¨ olldobler et al., 1999], but it was shown
with the help of Funahashi’s theorem, which is non-constructive as we have
already observed. In this section, we outline the ideas underlying the general
problem and also discuss different constructive approaches to it. But before
going into details, we need to answer the following questions.
• Why do we need to approximate operators such as the T P -operator?
• What does approximation mean in our context?
The first question is easily answered: even a single application of the T P
operator can lead to infinite results. For example, assume P is a program
containing the fact p(X). Applying the T P -operator once (to an arbitrary
interpretation) leads to a result containing infinitely many atoms, namely, all
p(X)-atoms for every X. In this simple example, we might be able to represent
this particular result in a finite way, but things might become arbitrarily
complex for other programs using the same or similar representations.
21
21 Indeed, the so-called rational models were developed to tackle this representational
problem for certain programs, see [Bornscheuer, 1996]. Unfortunately, there is no way to
compute an upper bound on the size of this rational representation, and hence it does not
give us any immediate advantages. Because we are not aware of any other finite representation, we will concentrate here on the standard representation using Herbrand interpretations.