Inductive Definition and Domain Theoretic Properties of Fully Abstract

TL;DR

This paper constructs fully abstract typed models for PCF and PCF^+ using level-by-level strategies, developing a new theory of computational strategies to establish domain-theoretic properties and model uniqueness.

Contribution

It introduces a novel level-by-level construction of fully abstract models for PCF and PCF^+, based on old notions of strategies, and develops a new theory of computational strategies for domain-theoretic analysis.

Findings

Models are sequentially complete and naturally continuous.

Models have definable universal functionals, ensuring uniqueness up to isomorphism.

Models are non-omega-complete but exhibit natural forms of completeness and algebraicity.

Abstract

A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF + "parallel conditional function"), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent non-deterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given level-by-level in the finite type hierarchy. To prove full abstraction and non-dcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF^+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are non-omega-complete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omega-completeness), "naturally" continuous (with respect to existing directed "pointwise", or "natural" lubs) and also "naturally" omega-algebraic and "naturally" bounded complete -- appropriate generalisation of the ordinary notions of domain theory to the case of non-dcpos.