Inductive Definition and Domain Theoretic Properties of Fully 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.

💡 Research Summary

The paper presents a novel construction of fully abstract typed models for the simply‑typed λ‑calculus with recursion (PCF) and its extension PCF⁺, which adds a parallel conditional operator. The construction proceeds level‑by‑level through the finite‑type hierarchy, using two concepts introduced by the author in the 1970s: sequential computational strategies and wittingly consistent nondeterministic strategies.

A sequential strategy formalises the way a program explores its arguments in a step‑by‑step fashion, demanding values only when needed. This captures the operational intuition of “call‑by‑need” without committing to a particular evaluation order. To accommodate the nondeterminism introduced by the parallel conditional, the author adds the notion of wittingly consistent strategies: whenever a nondeterministic choice is made, the strategy must guarantee that all possible branches remain mutually consistent, preventing contradictory behaviours.

The model is built inductively. At the base level (type N of natural numbers) the domain is the usual flat cpo, which is ω‑complete. For a higher‑order type σ→τ, the domain consists of all sequential strategies that, given a σ‑strategy as input, produce a τ‑strategy as output. Crucially, the construction guarantees the existence of “natural” least upper bounds (lub’s) for every directed set that can actually be generated by strategies. These natural lub’s are pointwise lub’s of the underlying numerical functions, and they exist even though the overall structure is not a dcpo in the classical sense. Consequently the model is not ω‑complete, but it is sequentially complete: every directed family of strategies that can be built sequentially has a lub in the model.

A central technical achievement is the definition of universal (surjective) functionals U_σ for each type σ. U_σ maps a numerical function f : ℕ→ℕ to a σ‑strategy that computes the same behaviour. Because U_σ is surjective, every element of the σ‑domain is representable by some numerical function, which yields a definable universality property. This universality immediately implies that any two fully abstract models of PCF (or PCF⁺) that support such universal functionals are isomorphic; thus the constructed model is unique up to isomorphism.

From a domain‑theoretic perspective the paper introduces four “natural” properties that replace the usual dcpo axioms:

1. Natural ω‑algebraicity – every element is the natural lub of a directed set of finitely generated (compact) strategies.
2. Natural bounded completeness – whenever a finite set has an upper bound, its least upper bound coincides with the natural lub.
3. Sequential completeness – directed families that arise from sequential construction possess lub’s, even though arbitrary directed families may not.
4. Natural continuity – functions between domains preserve natural lub’s (f(⋁_nat D) = ⋁_nat f