Sensible Intersection Type Theories

Finitary/static semantics in the form of intersection type assignments have become a paradigm for analysing the fine structure of all sorts of λ-models. The key step is the construction of a filter model isomorphic to a given λ-model. A property of great interest of filter λ-models is sensibility, i.e. the interpretation of all unsolvable terms is the least element. The flexibility of intersection type assignments derives from their parametrisation on intersection type theories. We construe intersection type theories as special meet-semilattices and show that appropriate morphisms, in the opposite category of meet-semilattices, preserve sensibility of the induced λ-models. Interestingly the set of saturated sets together with the set of λ-terms is such a meet-semilattice, thus showing that arguments based on Tait-Girards’s computability amount to the construction of a morphism. We characterise two classes of intersection type theories which induce sensible filter models. The first is non-effective while the second is effective and it amounts to the generalisation of Mendler’s criterion to intersection types and head normalising terms. The complete characterisation of sensible filter models however still escapes.