Relational Graph Models at Work

Abstract

We study the relational graph models that constitute a natural subclass ofrelational models of lambda-calculus. We prove that among the lambda-theoriesinduced by such models there exists a minimal one, and that the correspondingrelational graph model is very natural and easy to construct. We then studyrelational graph models that are fully abstract, in the sense that they capturesome observational equivalence between lambda-terms. We focus on the two mainobservational equivalences in the lambda-calculus, the theory H+ generated bytaking as observables the beta-normal forms, and H* generated by considering asobservables the head normal forms. On the one hand we introduce a notion oflambda-K\"onig model and prove that a relational graph model is fully abstractfor H+ if and only if it is extensional and lambda-K\"onig. On the other handwe show that the dual notion of hyperimmune model, together withextensionality, captures the full abstraction for H*