8 Eeective Cpos over Lambda Terms

In this paper, we have developed and analysed per models for recursive types, polymorphism, and subtyping. In developing our models, the emphasis has been on simplicity and on understanding the necessity of any conditions we impose on our structures. While similar structures have been given using the interpretation of constructive logic in a topos, the presentation of the models here is more direct and the analysis of their properties uses only simple recursion-theoretic arguments. Our main technical results fall into two parts. The rst set concern the properties of the model using partial equivalence relations over natural numbers , where types are coded by realizers of their supremum operators. We have established parametricity properties for an interpetation of polymorphism which is less uniform than \intersection", and coherence of the semantics of subtyping inspite of a non-uniform xed point operator. Our second set of results concern the comparison of uniformity of xed-point operators in GG odel number and lambda term models. We have shown the surprising property that no untyped xed point operator can be a xed point operator for any per that is a strict sub-quotient of the natural numbers, and established the correctness of an untyped xed point operator when we consider pers over lambda terms instead. problem of uniformly computing xed points of f: (R * S) !(R * S) for some class of pers R; S. A natural candidate for a uniform xed point operator is the index, Y , of the untyped call-by-value xed point operator (also given by the rst recursion theorem e.g., see Cut80]). Of course, untyped lambda calculus abounds with other xed point operators (c.f.Bar84]); the following deenition captures the essential property of any untyped xed point operator. Deenition13. A natural number f is an untyped xed point operator if for any n: N !N we have f n (N * N) n (f n). Lemma 14 (Failure of Untyped Fixed Points). Let f be any untyped xed point operator. Suppose that R; S are pers with :(k(R *S)k), for some k 2 N. not even realize a morphism from ((R * S) !(R * S)) to (R * S) and hence cannot realize a xed point operator for the per R * S. In particular, the index Y cannot be a realizer for the xed point operator on all R* S. The deenition of e, in the proof of Lemma 14, uses …