Logical Semantics for Stability

Type assignment systems for λ-calculus based on intersection types are a general framework for building models of λ-calculus (known as filter-models) which are useful tools for reasoning in a finitary way about the denotational interpretation of terms. Indeed the denotation of a term is the set of types derivable for it and a type is a “finite piece” of information on such a denotation. This approach to the λ-calculus semantics is called in the literature logical semantics, and it has been intensively studied in relation with λ-models in the Scott's domain setting. In this paper we define two intersection type assignment systems for λ-calculus, parametric with respect to a coherence relation between types. We prove that, when the instantiation of the parameter satisfies a given condition, our two type systems induce models of λ-calculus, that we call clique-models. Lastly we show that such systems give a logical characterization of two classes of models built on the category of Girard's coherence spaces and stable functions