218
Mathematical Aspects of Logic Programming Semantics
7.7 Some Extensions – The First-Order Case
So far, in this chapter we have described how certain methods developed
in earlier chapters give rise to approaches to the problem of integrating logic
programs and artificial neural networks. The key insight into this integration
is the observation that the two paradigms can be formally related by means of
functions: on the one hand, semantic operators for logic programs capture the
meaning of logic programs; on the other hand, the input-output function of
an artificial neural network completely characterizes its functional behaviour.
Approaches to neural-symbolic integration thus arise out of methods which
allow us to understand semantic operators as I/O functions of artificial neural
networks, and vice-versa.
Most of this chapter has focused on the single-step operator in logic programming which, via its fixed points, determines the supported model semantics of logic programs. However, in Section 7.6, we have just seen that some of
these methods carry over to other semantics, for propositional programs, via
the computation of F P . In this section, we will now consider the first-order
case and extensions of the approximation results we have established for T P to
F P and other semantic operators. At the same time, we briefly discuss further
alternative semantics as treated throughout the book and discuss conclusions
which can be drawn concerning neural-symbolic integration in general.
Our conceptual starting point is Theorem 7.5.3, which tells us that approximating networks exist if and only if the single-step operator is continuous in
the Cantor topology.
42 We can now use this result to leverage several new
results on the relationship between the supported model semantics and other
semantics in order to derive similar results for these other semantics.
In Section 5.4, we considered a very general family of semantic operators
and also examined the question of how one may characterize Cantor continuity
for them. The following result is thus an easy corollary of Theorem 5.4.7.
7.7.1 Theorem Let P be a program with a locally finite local consequence
operator T . Then T can be uniformly approximated by 3-layer feedforward
networks in the sense of Theorem 7.2.2.
42 We briefly remark that a result established in [Hornik et al., 1989], which states that
every measurable function can be approximated almost everywhere by a 3-layer sigmoidal
feedforward network, is not necessarily useful for our purposes. This is so despite the fact
that it was shown in Theorem 5.5.1 that many semantic operators, including the single-step
operator, are always measurable, and hence also the Gelfond–Lifschitz operator, see Theorem 6.2.4. However, it should be noted that the Cantor set is a set of (Lebesgue) measure zero
when viewed as a subspace of the reals. Thus, the result just quoted of [Hornik et al., 1989]
need not necessarily lead to useful approximation results for these operators. Indeed, such
approximations arising from the result of [Hornik et al., 1989] may fail to approximate the
operator in question at every point. Nevertheless, it remains to be investigated whether nonzero measures exist on the Cantor set, which yield useful approximations in conjunction with
the results of [Hornik et al., 1989].
Mathematical Aspects of Logic Programming Semantics
7.7 Some Extensions – The First-Order Case
So far, in this chapter we have described how certain methods developed
in earlier chapters give rise to approaches to the problem of integrating logic
programs and artificial neural networks. The key insight into this integration
is the observation that the two paradigms can be formally related by means of
functions: on the one hand, semantic operators for logic programs capture the
meaning of logic programs; on the other hand, the input-output function of
an artificial neural network completely characterizes its functional behaviour.
Approaches to neural-symbolic integration thus arise out of methods which
allow us to understand semantic operators as I/O functions of artificial neural
networks, and vice-versa.
Most of this chapter has focused on the single-step operator in logic programming which, via its fixed points, determines the supported model semantics of logic programs. However, in Section 7.6, we have just seen that some of
these methods carry over to other semantics, for propositional programs, via
the computation of F P . In this section, we will now consider the first-order
case and extensions of the approximation results we have established for T P to
F P and other semantic operators. At the same time, we briefly discuss further
alternative semantics as treated throughout the book and discuss conclusions
which can be drawn concerning neural-symbolic integration in general.
Our conceptual starting point is Theorem 7.5.3, which tells us that approximating networks exist if and only if the single-step operator is continuous in
the Cantor topology.
42 We can now use this result to leverage several new
results on the relationship between the supported model semantics and other
semantics in order to derive similar results for these other semantics.
In Section 5.4, we considered a very general family of semantic operators
and also examined the question of how one may characterize Cantor continuity
for them. The following result is thus an easy corollary of Theorem 5.4.7.
7.7.1 Theorem Let P be a program with a locally finite local consequence
operator T . Then T can be uniformly approximated by 3-layer feedforward
networks in the sense of Theorem 7.2.2.
42 We briefly remark that a result established in [Hornik et al., 1989], which states that
every measurable function can be approximated almost everywhere by a 3-layer sigmoidal
feedforward network, is not necessarily useful for our purposes. This is so despite the fact
that it was shown in Theorem 5.5.1 that many semantic operators, including the single-step
operator, are always measurable, and hence also the Gelfond–Lifschitz operator, see Theorem 6.2.4. However, it should be noted that the Cantor set is a set of (Lebesgue) measure zero
when viewed as a subspace of the reals. Thus, the result just quoted of [Hornik et al., 1989]
need not necessarily lead to useful approximation results for these operators. Indeed, such
approximations arising from the result of [Hornik et al., 1989] may fail to approximate the
operator in question at every point. Nevertheless, it remains to be investigated whether nonzero measures exist on the Cantor set, which yield useful approximations in conjunction with
the results of [Hornik et al., 1989].