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