New Results on Morris's Observational Theory: The Benefits of Separating the Inseparable

Working in the untyped lambda calculus, we study Morris’s λ-theory H+. Introduced in 1968, this is the original extensional theory of contextual equivalence. On the syntactic side, we show that this λ-theory validates the ω-rule, thus settling a long-standing open problem. On the semantic side, we provide sufficient and necessary conditions for relational graph models to be fully abstract for H+. We show that a relational graph model captures Morris’s observational preorder exactly when it is extensional and λ-König. Intuitively, a model is λ-König when every λ-definable tree has an infinite path which is witnessed by some element of the model. Both results follows from a weak separability property enjoyed by terms differing only because of some infinite η-expansion, which is proved through a refined version of the Böhm-out technique. 1998 ACM Subject Classification F.4.1 Mathematical Logic