137
Fixed-Point Theory for Generalized Metric Spaces
set programming systems which have been investigated with implementation
in mind, see [Lifschitz, 1999, Marek and Truszczy´ nski, 1999]. In addition, in
[Rounds and Zhang, 2001, Zhang and Rounds, 2001], Rounds and Zhang introduced a domain-theoretic framework for the study of the semantics of logic
programming, both procedural and non-procedural, including an abstract resolution rule, together with a treatment of negation, which is not negation as (finite) failure, however. [Hitzler, 2003a, Hitzler and Wendt, 2003, Hitzler, 2004,
Hitzler and Kr¨ otzsch, 2006] further expand on some aspects of the work of
Rounds and Zhang and in particular relate it to Formal Concept Analysis
[Ganter and Wille, 1999] and to answer set programming.
27
Of course, the monotonicity notions for multivalued mappings used mainly
in this chapter correspond to orderings encountered in power domains. In particular, this applies to Hoare montonicity and to Smyth monotonicity. With
this and the comments of the previous paragraph in mind, we note finally that
in Chapter 6 of [Zhang and Rounds, 2001], a treatment is given of the semantics of disjunctive logic programs (as considered here) with the same overall
objective as our own. The treatment is based on the Smyth powerdomain
again. One important feature of this power-domain approach is that by using
the right domain, the concept of multivalued function is avoided and continuity can always be taken to be Scott continuity. Thus, in conclusion, we note
that overall the developments just described appear to hold out, in particular,
the possibility of a domain-theoretic treatment of the declarative semantics
of negation in logic programming and therefore to bring logic programming
semantics more fully into the realm of domain theory, and vice-versa.
27 See Footnote 3 in the Introduction.
Fixed-Point Theory for Generalized Metric Spaces
set programming systems which have been investigated with implementation
in mind, see [Lifschitz, 1999, Marek and Truszczy´ nski, 1999]. In addition, in
[Rounds and Zhang, 2001, Zhang and Rounds, 2001], Rounds and Zhang introduced a domain-theoretic framework for the study of the semantics of logic
programming, both procedural and non-procedural, including an abstract resolution rule, together with a treatment of negation, which is not negation as (finite) failure, however. [Hitzler, 2003a, Hitzler and Wendt, 2003, Hitzler, 2004,
Hitzler and Kr¨ otzsch, 2006] further expand on some aspects of the work of
Rounds and Zhang and in particular relate it to Formal Concept Analysis
[Ganter and Wille, 1999] and to answer set programming.
27
Of course, the monotonicity notions for multivalued mappings used mainly
in this chapter correspond to orderings encountered in power domains. In particular, this applies to Hoare montonicity and to Smyth monotonicity. With
this and the comments of the previous paragraph in mind, we note finally that
in Chapter 6 of [Zhang and Rounds, 2001], a treatment is given of the semantics of disjunctive logic programs (as considered here) with the same overall
objective as our own. The treatment is based on the Smyth powerdomain
again. One important feature of this power-domain approach is that by using
the right domain, the concept of multivalued function is avoided and continuity can always be taken to be Scott continuity. Thus, in conclusion, we note
that overall the developments just described appear to hold out, in particular,
the possibility of a domain-theoretic treatment of the declarative semantics
of negation in logic programming and therefore to bring logic programming
semantics more fully into the realm of domain theory, and vice-versa.
27 See Footnote 3 in the Introduction.