Full Abstraction and the Context Lemma

Abstract

It is impossible to add a combinator to PCF to achieve full abstraction for models such as Berry’s stable domains in a way analogous to the addition of the “parallel-or” combinator that achieves full abstraction for the familiar complete partial order (cpo) model. In particular, we define a general notion of rewriting system of the kind used for evaluating simply typed $\lambda $-terms in Scott’s PCF. Any simply typed $\lambda $-calculus with such a “PCF-like” rewriting semantics is shown necessarily to satisfy Miler’s Context Lemma. A simple argument demonstrates that any denotational semantics that is adequate for PCF, and in which certain simple Boolean functionals exist, cannot be fully abstract for any extension of PCF satisfying the Context Lemma. An immediate corollary is that stable domains cannot be fully abstract for any extension of PCF definable by PCF-like rules.