An Information-system Representation of the Smyth Powerdomain

This paper provides a representation of the Smyth powerdomain as information systems. A new notion of ideal elements, called disjunctive states, is introduced. Disjunctive states are built from clauses over the token set of the underlying information system in order to represent disjunctive information. At the heart of this representation is a hyperresolution rule, whose completeness hinges upon a crucial combinatorial lemma for the proper book-keeping of intermediate clauses. Our main representation result uses the Hooman-Mislove Theorem to establish an order-isomorphism between disjunc-tive states and compact, saturated sets. As an application, we provide a couple of speciic Smyth powerdomain examples that are useful for disjunctive logic programming. Moreover, the notion of disjunctive state is immediately applicable to sequent structures, or nondeterministic information systems. We show that the hyperresolution rule is sound and complete for sequent structures as well. 1. Introduction In domain theory one is concerned with appropriate mathematical spaces in which to present the denotational semantics of programming languages. The re-cursive, self-applicative nature of programming constructs imposes structural properties on these spaces. The existence of mathematical spaces that satisfy some seemingly stringent requirements was rst demonstrated by Scott in the late 60's. The program started by him has now become a mature area which is theoretically elegant (see, for example, Abramsky and Jung 1]) and practically useful (see, for example, Amadio and Curien 2], Gunter 5], Winskel 18]). This paper provides a consideration of the Smyth powerdomain as a mathematical space for disjunctive logic programming. Speciically, we introduce an information-system representation of the Smyth powerdomain and use this representation as guidance to develop a xed-point semantics for disjunctive logic programming. This concrete representation provides a link between the abstract clausal logic on algebraic domains 13] and the standard logic programming semantics .