Topological characterizations for logic programming semantics

In terms of the arithmetic hierarchy, the complexity of the set of minimal models, the set of stable models, and the set of supported models of a propositional general logic program has previously been described. However, not every set of interpretations of this level of complexity is obtained as such a set. The main contribution of this paper is to characterize among the sets of appropriate interpretations for the language those which are the minimal, stable, and supported model classes of programs. This is accomplished by viewing the interpretations as points in a topological space; the subsets of the space corresponding to classes of a given sort are then found to be distinguished by particular topological properties. The study involves both the Cantor topology familiar from descriptive set theory and the less well-known inverse Scott topology, defined originally in domain theory in connection with representing information about computations. Closely related to the topological characterizations are results of two other sorts: logical descriptions of the classes in terms of formulas extracted from the programs (known previously for supported and stable model classes) and normal forms for programs representing given classes (known previously for stable model classes). We obtain a description of the first sort for minimal model classes and provide normal form representations for minimal and supported model classes. As an application of compactness, we obtain characterization results for the stable and supported model classes of programs with certain finiteness properties. Finally we determine how the classes of intended models are affected when classical negation is introduced. We include numerous examples to clarify the relations among different properties and classes. We also include examples of calculations which can be performed by these methods.