Chapter 8
Final Thoughts
In this book, we have provided a comprehensive treatment of logic programming semantics from the perspective of fixed-point semantics. In doing so, we
have covered a lot of material which also relates to other areas of interest
outside the realm of logic programming as such. In this final chapter, we discuss contributions to and relationships between the content of this book and a
rather diverse mix of topics, ranging from foundations of computing via artificial intelligence to cognitive science. We do so with the usual understanding
that the impact of foundational research is more often than not indirect in
nature in providing results, methods, and insights, which can be carried forward by research communities at large until a critical mass is reached, thereby
enabling significant or even major advances to take place.
8.1 Foundations of Programming Semantics
The classical semantic analysis of programs in the sense of denotational
semantics is based on monotonic, order-continuous operators, via their least
fixed points using Theorem 1.1.9 or Theorem 1.1.10. This approach, however,
fails for paradigms where the semantics is expressed by fixed points of operators which are not monotonic in general. In particular, it fails for logic
programming in several of its variants, as studied throughout this book.
By developing methods for the fixed-point semantic analysis of programs
with non-monotonic semantic operators, we therefore widen the scope of applicability of fixed-point semantics. In particular, we provide sufficiency conditions for the existence of fixed points (Chapter 4) and show how they can
be applied to various semantics based on non-monotonic operators (Sections
5.1 and 5.4 and Chapter 6).
It seems evident that these methods should carry over to other such
paradigms. However, a limitation of some of the work presented in this book
is that certain of the fixed-point theorems provided in Chapter 4 always guarantee the existence of a unique fixed point, if there is a fixed point at all,
thus rendering the theorems in question of limited applicability to paradigms
(or programs) where multiple fixed points are the norm. The latter situation
221
Final Thoughts
In this book, we have provided a comprehensive treatment of logic programming semantics from the perspective of fixed-point semantics. In doing so, we
have covered a lot of material which also relates to other areas of interest
outside the realm of logic programming as such. In this final chapter, we discuss contributions to and relationships between the content of this book and a
rather diverse mix of topics, ranging from foundations of computing via artificial intelligence to cognitive science. We do so with the usual understanding
that the impact of foundational research is more often than not indirect in
nature in providing results, methods, and insights, which can be carried forward by research communities at large until a critical mass is reached, thereby
enabling significant or even major advances to take place.
8.1 Foundations of Programming Semantics
The classical semantic analysis of programs in the sense of denotational
semantics is based on monotonic, order-continuous operators, via their least
fixed points using Theorem 1.1.9 or Theorem 1.1.10. This approach, however,
fails for paradigms where the semantics is expressed by fixed points of operators which are not monotonic in general. In particular, it fails for logic
programming in several of its variants, as studied throughout this book.
By developing methods for the fixed-point semantic analysis of programs
with non-monotonic semantic operators, we therefore widen the scope of applicability of fixed-point semantics. In particular, we provide sufficiency conditions for the existence of fixed points (Chapter 4) and show how they can
be applied to various semantics based on non-monotonic operators (Sections
5.1 and 5.4 and Chapter 6).
It seems evident that these methods should carry over to other such
paradigms. However, a limitation of some of the work presented in this book
is that certain of the fixed-point theorems provided in Chapter 4 always guarantee the existence of a unique fixed point, if there is a fixed point at all,
thus rendering the theorems in question of limited applicability to paradigms
(or programs) where multiple fixed points are the norm. The latter situation
221