# Every λ-Term is Meaningful for the Infinitary Relational Model

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Inria Rennes – Bretagne Atlantique , LS2N - Laboratoire des Sciences du Numérique de Nantes
Abstract : Infinite types and formulas are known to have really curious and unsound behaviors. For instance, they allow to type $Ω$, the auto-autoapplication and they thus do not ensure any form of normal-ization/productivity. Moreover, in most infinitary frameworks, it is not difficult to define a type R that can be assigned to every $λ$-term. However, these observations do not say much about what coinductive (i.e. infinitary) type grammars are able to provide: it is for instance very difficult to know what types (besides R) can be assigned to a given term in this setting. We begin with a discussion on the expressivity of different forms of infinite types. Then, using the resource-awareness of sequential intersection types (system S) and tracking, we prove that infinite types are able to characterize the arity of every λ-terms and that, in the infinitary extension of the relational model, every term has a " meaning " i.e. a non-empty denotation. From the technical point of view, we must deal with the total lack of guarantee of productivity for typable terms: we do so by importing methods inspired by first order model theory.
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Type de document :
Communication dans un congrès
Anuj Dawar, Erich Grädel. LICS 2018 - Thirty-Third Annual ACM/IEEE Symposium on Logic in Computer Science, Jul 2018, Oxford, United Kingdom. ACM, pp.1-26, 2018, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018. 〈10.1145/nnnnnnn.nnnnnnn〉
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https://hal.archives-ouvertes.fr/hal-01840744
Contributeur : Pierre Vial <>
Soumis le : jeudi 26 juillet 2018 - 14:53:04
Dernière modification le : vendredi 1 mars 2019 - 09:57:22
Document(s) archivé(s) le : samedi 27 octobre 2018 - 12:11:11

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hal-inf-rel-mod-conf.pdf
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### Citation

Pierre Vial. Every λ-Term is Meaningful for the Infinitary Relational Model. Anuj Dawar, Erich Grädel. LICS 2018 - Thirty-Third Annual ACM/IEEE Symposium on Logic in Computer Science, Jul 2018, Oxford, United Kingdom. ACM, pp.1-26, 2018, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018. 〈10.1145/nnnnnnn.nnnnnnn〉. 〈hal-01840744〉

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