226
Mathematical Aspects of Logic Programming Semantics
opposed to a cognitive science motivation as discussed in Section 8.6), that
is, by the idea of combining logic programming and artificial neural networks
in such a way that the best of both worlds – declarativeness, trainability,
robustness, and reasoning capabilities – is retained.
Indeed, this effort has paid off, and while we provide only the theoretical
underpinnings in Chapter 7, we are indeed able to show that a declarative,
trainable, robust, and reasonable system can be developed on these grounds,
14
although it has to be said that the advance remains conceptual in nature
because the system is severely limited in terms of the size of the knowledge
base involved. Nevertheless, it is to date one of the two reported systems with
these capabilities.
15
Significant further advances on this front, in particular with respect to
the integration of learning and reasoning, would be highly appreciated in
practice.
16
8.8 Topology, Programming, and Artificial Intelligence
It has been argued that there is a strong relationship between topological dynamics (chaos theory), logic programming, neural networks, and other
paradigms, and in particular this is so in the context of emergent behaviour
as represented by cellular automata, say.
17 Indeed, from a bird’s eye perspective each seems to be capable of being mapped onto the others. At the same
time, the study in any one of these paradigms seems to pose the same sort of
obstacles found in the others, particularly is this so in relation to the handling
of chaotic dynamics and emergence.
Some of the work in this book contributes to this discussion, especially with
respect to topological dynamics, logic programming, and neural networks, as
discussed in Section 7.5.
18 Obviously, this is only a small stepping stone in
the pursuit of these issues which, once fully understood, will provide a major
14 See [Bader et al., 2007, Bader et al., 2008, Bader, 2009] for details.
15 The approach in [Gust et al., 2007] achieves similar results with entirely different methods.
16 For a discussion of the Semantic Web (see Section 8.4) as a potential test case for neuralsymbolic integration, see [Hitzler et al., 2005]. For a general discussion of the need for the
integration of learning and reasoning for Semantic Web applications, see [Hitzler, 2009,
Hitzler and van Harmelen, 2010].
17 See, for example, [Blair et al., 1997a, Blair et al., 1999]
18 In [Bader and Hitzler, 2004], it was shown that there is indeed a tight relationship
between logic programs and fractals in the sense in which they arise as attractors of iterated
function systems.
Mathematical Aspects of Logic Programming Semantics
opposed to a cognitive science motivation as discussed in Section 8.6), that
is, by the idea of combining logic programming and artificial neural networks
in such a way that the best of both worlds – declarativeness, trainability,
robustness, and reasoning capabilities – is retained.
Indeed, this effort has paid off, and while we provide only the theoretical
underpinnings in Chapter 7, we are indeed able to show that a declarative,
trainable, robust, and reasonable system can be developed on these grounds,
14
although it has to be said that the advance remains conceptual in nature
because the system is severely limited in terms of the size of the knowledge
base involved. Nevertheless, it is to date one of the two reported systems with
these capabilities.
15
Significant further advances on this front, in particular with respect to
the integration of learning and reasoning, would be highly appreciated in
practice.
16
8.8 Topology, Programming, and Artificial Intelligence
It has been argued that there is a strong relationship between topological dynamics (chaos theory), logic programming, neural networks, and other
paradigms, and in particular this is so in the context of emergent behaviour
as represented by cellular automata, say.
17 Indeed, from a bird’s eye perspective each seems to be capable of being mapped onto the others. At the same
time, the study in any one of these paradigms seems to pose the same sort of
obstacles found in the others, particularly is this so in relation to the handling
of chaotic dynamics and emergence.
Some of the work in this book contributes to this discussion, especially with
respect to topological dynamics, logic programming, and neural networks, as
discussed in Section 7.5.
18 Obviously, this is only a small stepping stone in
the pursuit of these issues which, once fully understood, will provide a major
14 See [Bader et al., 2007, Bader et al., 2008, Bader, 2009] for details.
15 The approach in [Gust et al., 2007] achieves similar results with entirely different methods.
16 For a discussion of the Semantic Web (see Section 8.4) as a potential test case for neuralsymbolic integration, see [Hitzler et al., 2005]. For a general discussion of the need for the
integration of learning and reasoning for Semantic Web applications, see [Hitzler, 2009,
Hitzler and van Harmelen, 2010].
17 See, for example, [Blair et al., 1997a, Blair et al., 1999]
18 In [Bader and Hitzler, 2004], it was shown that there is indeed a tight relationship
between logic programs and fractals in the sense in which they arise as attractors of iterated
function systems.