International audienceWe define the class of relational graph models and study the induced order-and equational-theories. Using the Taylor expansion, we show that all λ-terms with the same Böhm tree are equated in any relational graph model. If the model is moreover extensional and satisfies a technical condition, then its order-theory coincides with Morris's observational pre-order. Finally, we introduce an extensional version of the Taylor expansion, then prove that two λ-terms have the same extensional Taylor expansion exactly when they are equivalent in Morris's sense

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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 λ-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 * . LOGICAL METHODS lI N COMPUTER SCIENCE