Introduction
xxvii
known Gelfond–Lifschitz operator GL P and the fixpoint completion fix(P ) for
any normal logic program P by deriving the identity GL P (I) = T fix(P ) (I), for
any two-valued interpretation I, see Theorem 6.1.4. This will make it a simple
and routine matter to prove many facts about GL P , and hence about the stable model, from properties of the single-step operator, including the derivation
of continuity properties of GL P . Our second objective in Chapter 6 is to revisit stratification and the perfect model and to present an iterative process for
obtaining the perfect model for locally stratified normal logic programs. This
approach involves careful control of negation in order to produce monotonic
increasing sequences by means of non-monotonic operators and is interesting
for the insight it gives into the structure of the perfect model. In Chapter 7,
we apply the topological and analytical tools developed earlier in order to
discuss logic programming in the context of dynamical systems and artificial
neural networks with a view, in particular, to presenting a detailed account of
these methods in the foundations of neural-symbolic integration. Specifically,
in Chapter 7, we consider the computation by artificial neural networks of
various semantic operators associated with normal logic programs. We view
this as a means of integrating these two computing paradigms because both
can be represented by functions: the semantic operator on the one hand and
the I/O function of the neural network on the other. In fact, exact computation of semantic operators is only possible in the case of propositional normal
logic programs. In the case of first-order programs, approximation methods
are required, and this is where analytical and topological methods make their
entrance. Indeed, it turns out that continuity of a semantic operator in the
Cantor topology is a necessary and sufficient condition for this approximation
process to work, see Theorem 7.5.3. This observation is yet further motivation for studying the Cantor topology, and hence Chapter 7 represents an
important application of analytical ideas in logic programming semantics. In
Chapter 8, we give a brief discussion of further possible applications of our
results and future directions for research involving the methods and results
of this book. In particular, we discuss possible future work in the context
of the foundations of program semantics, quantitative domain theory, fixedpoint theory, the Semantic Web, and neural-symbolic integration, among other
things. In the Appendix, we bring together a summary of those facts from the
theory of ordinals and general topology which will be needed at various points
in our investigations, but are not developed in the main body of the text; its
inclusion makes our treatment essentially self-contained. In particular, the results of Chapter 3 together with those of the Appendix give a treatment of
the Scott topology in terms of convergence.
Finally, on a point of convention, we note that the symbol • will be employed as an end marker in two ways in the body of the text. First, it will
be used to indicate the end of every proof. Second, it will be used on a few
occasions to mark clearly the end of any statement (theorem, proposition,
definition, remark, example, program, etc.), where the end of that statement
might otherwise be unclear.
xxvii
known Gelfond–Lifschitz operator GL P and the fixpoint completion fix(P ) for
any normal logic program P by deriving the identity GL P (I) = T fix(P ) (I), for
any two-valued interpretation I, see Theorem 6.1.4. This will make it a simple
and routine matter to prove many facts about GL P , and hence about the stable model, from properties of the single-step operator, including the derivation
of continuity properties of GL P . Our second objective in Chapter 6 is to revisit stratification and the perfect model and to present an iterative process for
obtaining the perfect model for locally stratified normal logic programs. This
approach involves careful control of negation in order to produce monotonic
increasing sequences by means of non-monotonic operators and is interesting
for the insight it gives into the structure of the perfect model. In Chapter 7,
we apply the topological and analytical tools developed earlier in order to
discuss logic programming in the context of dynamical systems and artificial
neural networks with a view, in particular, to presenting a detailed account of
these methods in the foundations of neural-symbolic integration. Specifically,
in Chapter 7, we consider the computation by artificial neural networks of
various semantic operators associated with normal logic programs. We view
this as a means of integrating these two computing paradigms because both
can be represented by functions: the semantic operator on the one hand and
the I/O function of the neural network on the other. In fact, exact computation of semantic operators is only possible in the case of propositional normal
logic programs. In the case of first-order programs, approximation methods
are required, and this is where analytical and topological methods make their
entrance. Indeed, it turns out that continuity of a semantic operator in the
Cantor topology is a necessary and sufficient condition for this approximation
process to work, see Theorem 7.5.3. This observation is yet further motivation for studying the Cantor topology, and hence Chapter 7 represents an
important application of analytical ideas in logic programming semantics. In
Chapter 8, we give a brief discussion of further possible applications of our
results and future directions for research involving the methods and results
of this book. In particular, we discuss possible future work in the context
of the foundations of program semantics, quantitative domain theory, fixedpoint theory, the Semantic Web, and neural-symbolic integration, among other
things. In the Appendix, we bring together a summary of those facts from the
theory of ordinals and general topology which will be needed at various points
in our investigations, but are not developed in the main body of the text; its
inclusion makes our treatment essentially self-contained. In particular, the results of Chapter 3 together with those of the Appendix give a treatment of
the Scott topology in terms of convergence.
Finally, on a point of convention, we note that the symbol • will be employed as an end marker in two ways in the body of the text. First, it will
be used to indicate the end of every proof. Second, it will be used on a few
occasions to mark clearly the end of any statement (theorem, proposition,
definition, remark, example, program, etc.), where the end of that statement
might otherwise be unclear.