223
Final Thoughts
driven work of quantitative domain theory, as discussed herein, relates to
theory-driven advances in the same topic, which were developed in parallel.
5
8.3 Fixed-Point Theorems for Generalized Metric Spaces
Fixed-point theorems have a rightful place in the core arsenal of mathematical tools applicable to theoretical computer science, with many applications
outside this realm of course. The Banach contraction mapping theorem, Theorem 4.2.3, which is the starting point for many of the investigations in this
book, is one of the most fundamental of these theorems.
In this book, we have contributed to the study of fixed-point theory by exploring generalized metrics and thereby providing a compilation of extensions
of the Banach theorem, together with results concerning their relationships
with order-theoretic fixed-point theorems (Chapter 4). We furthermore provide evidence of the usefulness of these theorems by applying them, throughout
the book, to the study of the semantics of logic programs.
Another theme which has not been taken up here, and again is scope for
further work, is a systematic investigation of the extent to which the Banach
theorem, and its relatives, remains valid with respect to generalized distance
functions under weaker and weaker conditions. Specifically, how weak can
the ambient spaces be and still support a reasonable version of the Banach
theorem?
8.4 The Foundations of Knowledge Representation and
Reasoning
Knowledge Representation and Reasoning (KR) is one of the classical
branches of Artificial Intelligence. Currently, it is experiencing massive renewed interest due to the advent of the Semantic Web.
6 In a nutshell, the Semantic Web strives to improve the World Wide Web by making Web content
machine-understandable, and it does so by using KR methods, more precisely,
by endowing Web content with additional meta-content in the form of knowledge bases (so-called ontologies), which describe the content in a logic-based
format.
Several KR languages have been developed and standardized by the World
5 The papers [Waszkiewicz, 2002, Waszkiewicz, 2003, Waszkiewicz, 2006, Kr¨ otzsch, 2006,
K¨ unzi et al., 2006, K¨ unzi and Kivuvu, 2008], for example, may be consulted.
6 See [Hitzler et al., 2009b] for an introductory textbook.
Final Thoughts
driven work of quantitative domain theory, as discussed herein, relates to
theory-driven advances in the same topic, which were developed in parallel.
5
8.3 Fixed-Point Theorems for Generalized Metric Spaces
Fixed-point theorems have a rightful place in the core arsenal of mathematical tools applicable to theoretical computer science, with many applications
outside this realm of course. The Banach contraction mapping theorem, Theorem 4.2.3, which is the starting point for many of the investigations in this
book, is one of the most fundamental of these theorems.
In this book, we have contributed to the study of fixed-point theory by exploring generalized metrics and thereby providing a compilation of extensions
of the Banach theorem, together with results concerning their relationships
with order-theoretic fixed-point theorems (Chapter 4). We furthermore provide evidence of the usefulness of these theorems by applying them, throughout
the book, to the study of the semantics of logic programs.
Another theme which has not been taken up here, and again is scope for
further work, is a systematic investigation of the extent to which the Banach
theorem, and its relatives, remains valid with respect to generalized distance
functions under weaker and weaker conditions. Specifically, how weak can
the ambient spaces be and still support a reasonable version of the Banach
theorem?
8.4 The Foundations of Knowledge Representation and
Reasoning
Knowledge Representation and Reasoning (KR) is one of the classical
branches of Artificial Intelligence. Currently, it is experiencing massive renewed interest due to the advent of the Semantic Web.
6 In a nutshell, the Semantic Web strives to improve the World Wide Web by making Web content
machine-understandable, and it does so by using KR methods, more precisely,
by endowing Web content with additional meta-content in the form of knowledge bases (so-called ontologies), which describe the content in a logic-based
format.
Several KR languages have been developed and standardized by the World
5 The papers [Waszkiewicz, 2002, Waszkiewicz, 2003, Waszkiewicz, 2006, Kr¨ otzsch, 2006,
K¨ unzi et al., 2006, K¨ unzi and Kivuvu, 2008], for example, may be consulted.
6 See [Hitzler et al., 2009b] for an introductory textbook.