225
Final Thoughts
It still remains to be seen, however, how far this approach can be carried,
12
and whether or not it is possible to establish a meta-theory which goes beyond
mere characterization.
13
8.6 Symbolic and Subsymbolic Representations
How to overcome the gap between symbolic and subsymbolic representations, and how to integrate them in an efficient and effective manner, is
a topic of growing interdisciplinary importance. It is driven by advances in
neuroimaging, which call for the modelling of findings in neuroscience on a
higher and higher level of abstraction, and by the search in Cognitive Science
for suitable cognitive architectures to model complex behaviour. By symbolic
we mean, of course, knowledge representation formalisms based on logic or
similar algebraic structures, while the term subsymbolic refers to paradigms
such as artificial neural networks, where knowledge is not represented in a
crisp, declarative way.
The topology-driven view of logic programming semantics which we pursue
herein indirectly embraces this theme by providing a conceptual bridge between the discrete (symbolic) world of logic and the continuous (subsymbolic)
world of topology and analysis on the reals.
While, originally, we developed this point of view purely for the purposes of
analyzing logic programs and in order to advance quantitative domain theory,
it bears, at least conceptually, on the symbolic/subsymbolic issue. However, we
have not pursued this in any structured manner, apart from developing neuralsymbolic integration (Chapter 7), albeit with a different initial motivation
(see Section 8.7). The question remains open to what extent our insights can
contribute to the larger quest.
8.7 Neural-Symbolic Integration
Our work on neural-symbolic integration started as a straightforward application of our topological approach to logic programming semantics. The
pursuit (Chapter 7) was then driven mainly by an engineering motivation (as
12 Disjunctive well-founded semantics were compared using this approach in the paper
[Knorr and Hitzler, 2007], but only with limited success since the characterizations became
rather complicated.
13 In [Cherchago et al., 2007], for example, level mappings were used to study decidability
properties.
Final Thoughts
It still remains to be seen, however, how far this approach can be carried,
12
and whether or not it is possible to establish a meta-theory which goes beyond
mere characterization.
13
8.6 Symbolic and Subsymbolic Representations
How to overcome the gap between symbolic and subsymbolic representations, and how to integrate them in an efficient and effective manner, is
a topic of growing interdisciplinary importance. It is driven by advances in
neuroimaging, which call for the modelling of findings in neuroscience on a
higher and higher level of abstraction, and by the search in Cognitive Science
for suitable cognitive architectures to model complex behaviour. By symbolic
we mean, of course, knowledge representation formalisms based on logic or
similar algebraic structures, while the term subsymbolic refers to paradigms
such as artificial neural networks, where knowledge is not represented in a
crisp, declarative way.
The topology-driven view of logic programming semantics which we pursue
herein indirectly embraces this theme by providing a conceptual bridge between the discrete (symbolic) world of logic and the continuous (subsymbolic)
world of topology and analysis on the reals.
While, originally, we developed this point of view purely for the purposes of
analyzing logic programs and in order to advance quantitative domain theory,
it bears, at least conceptually, on the symbolic/subsymbolic issue. However, we
have not pursued this in any structured manner, apart from developing neuralsymbolic integration (Chapter 7), albeit with a different initial motivation
(see Section 8.7). The question remains open to what extent our insights can
contribute to the larger quest.
8.7 Neural-Symbolic Integration
Our work on neural-symbolic integration started as a straightforward application of our topological approach to logic programming semantics. The
pursuit (Chapter 7) was then driven mainly by an engineering motivation (as
12 Disjunctive well-founded semantics were compared using this approach in the paper
[Knorr and Hitzler, 2007], but only with limited success since the characterizations became
rather complicated.
13 In [Cherchago et al., 2007], for example, level mappings were used to study decidability
properties.