Introduction
xxv
neural-symbolic integration. Such non-classical models of computation are of
great interest generally in present times and may contain both continuous
and discrete components, especially those inspired by physical phenomena.
As such, their study will almost certainly require techniques appropriate to
both their continuous elements and to their discrete elements and may well
be of the sort developed here.
It should be noted that other authors have, to a greater or lesser extent,
employed mathematical analysis in the context of logic programming semantics. Their work is complementary to what we present here, and we briefly
discuss some of it and its relationship with ours next and in more detail in the
body of the text. For example, some of the recent work of Howard Blair and
several of his colleagues on logic programming semantics is much concerned
with the interaction between the continuous and the discrete, and it makes
use of ideas from dynamical systems, convergence spaces, and automata theory to model hybrid systems. We consider this work further in Chapter 3.
We mention also the work of Sibylla Prieß-Crampe and Paulo Ribenboim on
the role of generalized ultrametrics in fixed-point theory in the context of
logic programmimg semantics. They discuss both single-valued and multivalued mappings in this context, and we consider their results in considerable
detail in Chapter 4 and some of their applications in Chapter 5. In addition,
we also include in Chapter 4 a discussion of recent work of Umberto Straccia,
Manuel Ojeda-Aciego, and Carlos Dam´ asio on multivalued mappings in the
context of semantics and the relationship between their work and ours. Finally, we discuss in Chapter 4 also the extensive work of William Rounds and
Guo-Qiang Zhang on the use of domain theory as a theoretical foundation
for logic programming, both from the point of view of procedural aspects and
from the point of view of semantics.
Summarizing the chapters, Chapter 1 contains, in fairly condensed form,
the preliminaries from order theory, domain theory, and logic which we will
employ throughout this book. In addition, we present two well-known fixedpoint theorems, based on order, which are fundamental in applications to
semantics. The next chapter, Chapter 2, introduces logic programs and the
most important ways of assigning semantic operators and declarative semantics to them. The manner in which the material is presented is rather novel and
employs the syntactic notion of level mapping, defined in Chapter 2. Indeed,
we make several different applications of level mappings in our discussions,
and they play a unifying role in several places in the course of developing our
main themes. For example, their use in Chapter 2 provides a uniform and
comprehensive treatment of all of the important different semantics known in
the subject, including those associated with the supported, stable, and wellfounded models mentioned earlier.
7 Sets of interpretations are important in
that they are, among other things, the carrier sets for the various semantic
7 The uniform characterizations by means of level mappings which will be given in Chapter 2 are due mainly to [Hitzler and Wendt, 2002].
xxv
neural-symbolic integration. Such non-classical models of computation are of
great interest generally in present times and may contain both continuous
and discrete components, especially those inspired by physical phenomena.
As such, their study will almost certainly require techniques appropriate to
both their continuous elements and to their discrete elements and may well
be of the sort developed here.
It should be noted that other authors have, to a greater or lesser extent,
employed mathematical analysis in the context of logic programming semantics. Their work is complementary to what we present here, and we briefly
discuss some of it and its relationship with ours next and in more detail in the
body of the text. For example, some of the recent work of Howard Blair and
several of his colleagues on logic programming semantics is much concerned
with the interaction between the continuous and the discrete, and it makes
use of ideas from dynamical systems, convergence spaces, and automata theory to model hybrid systems. We consider this work further in Chapter 3.
We mention also the work of Sibylla Prieß-Crampe and Paulo Ribenboim on
the role of generalized ultrametrics in fixed-point theory in the context of
logic programmimg semantics. They discuss both single-valued and multivalued mappings in this context, and we consider their results in considerable
detail in Chapter 4 and some of their applications in Chapter 5. In addition,
we also include in Chapter 4 a discussion of recent work of Umberto Straccia,
Manuel Ojeda-Aciego, and Carlos Dam´ asio on multivalued mappings in the
context of semantics and the relationship between their work and ours. Finally, we discuss in Chapter 4 also the extensive work of William Rounds and
Guo-Qiang Zhang on the use of domain theory as a theoretical foundation
for logic programming, both from the point of view of procedural aspects and
from the point of view of semantics.
Summarizing the chapters, Chapter 1 contains, in fairly condensed form,
the preliminaries from order theory, domain theory, and logic which we will
employ throughout this book. In addition, we present two well-known fixedpoint theorems, based on order, which are fundamental in applications to
semantics. The next chapter, Chapter 2, introduces logic programs and the
most important ways of assigning semantic operators and declarative semantics to them. The manner in which the material is presented is rather novel and
employs the syntactic notion of level mapping, defined in Chapter 2. Indeed,
we make several different applications of level mappings in our discussions,
and they play a unifying role in several places in the course of developing our
main themes. For example, their use in Chapter 2 provides a uniform and
comprehensive treatment of all of the important different semantics known in
the subject, including those associated with the supported, stable, and wellfounded models mentioned earlier.
7 Sets of interpretations are important in
that they are, among other things, the carrier sets for the various semantic
7 The uniform characterizations by means of level mappings which will be given in Chapter 2 are due mainly to [Hitzler and Wendt, 2002].