Relational Graph Models at Work

We study the relational graph models that constitute a natural subclass of relational models of λ-calculus. We prove that among the λ-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between λ-terms. We focus on the two main observational equivalences in the λ-calculus, the theory H + generated by taking as observables the β-normal forms, and H * generated by considering as observables the head normal forms. On the one hand we introduce a notion of λ-König model and prove that a relational graph model is fully abstract for H + if and only if it is extensional and λ-König. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H * .