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Modèles de Graphe Relationnels et Observabilité à la Morris : recherches sémantiques sensibles aux ressources sur le λ-calcul non typé

Abstract : This thesis is a contribution to the study of Church’s untyped λ-calculus, a term rewritingsystem having the β-reduction (the formal counterpart of the idea of execution of programs) asmain rule. The focus is on denotational semantics, namely the investigation of mathematical models of the λ-calculus giving the same denotation to β-convertible λ-terms. We investigate relational semantics, a resource-sensitive semantics interpreting λ-terms as relations,with their inputs grouped together in multisets. We define a large class of relational models,called relational graph models (rgm’s), and we study them in a type/proof-theoretical way, using some non-idempotent intersection type systems. Firstly, we find the minimal and maximal λ-theories (equational theories extending -conversion) represented by the class.Then we use rgm’s to solve the full abstraction problem for Morris’s observational λ-theory,the contextual equivalence of programs that one gets by taking the β-normal forms asobservable outputs. We solve the problem in different ways. Through a type-theoretical characterization of β-normalizability, we find infinitely many fully abstract rgm’s, that wecall uniformly bottomless.We then give an exhaustive answer to the problem, by showing thatan rgm is fully abstract for Morris’s observability if and only if it is extensional (a model of ŋ-conversion) and λ-König. Intuitively an rgm is λ-König when every infinite computable tree has an infinite branch witnessed by some type of the model, where the witnessing is a property of non-well-foundedness on the type.
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Domenico Ruoppolo. Modèles de Graphe Relationnels et Observabilité à la Morris : recherches sémantiques sensibles aux ressources sur le λ-calcul non typé. Ordinateur et société [cs.CY]. Université Sorbonne Paris Cité, 2016. Français. ⟨NNT : 2016USPCD069⟩. ⟨tel-02132865⟩

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