We show how to characterize compositionally a number of evaluation properties of λ-terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalization, normalization, head normalization, and weak head normalization. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterization results are new, to our knowledge, or else they streamline, strengthen, or generalize earlier results in the literature. The completeness parts of the characterizations are proved uniformly for all the properties, using a set-theoretical semantics of intersection types over suitable kinds of stable sets. This technique generalizes Krivine 's and Mitchell's methods for strong normalization to other evaluation properties.

We illustrate the use of intersection types as a tool for synthesizing -models which exhibit special purpose features. We focus on semantical proofs of easiness. This allows us to prove that the class of -theories induced by graph models is strictly included in the class of -theories induced by non-extensional lter models.

Invariance of interpretation by #-conversion is one of the minimal requirements for any standard model for the #-calculus. With the intersection type systems being a general framework for the study of semantic domains for the #-calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection type assignment systems.

We show how to characterise compositionally a number of evaluation properties of λ-terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterisation results are new, to our knowledge, or else they streamline, strengthen, or generalise earlier results in the literature. The completeness parts of the characterisations are proved uniformly for all the properties, using a set-theoretical semantics of intersection types over suitable kinds of stable sets. This technique generalises Krivine's and Mitchell's methods for strong normalisation to other evaluation properties.

3.1 Presentation of the λ-Calculus..................... 5