Fabio Alessi

Intersection Types and Computational Rules

Abstract The invariance of the meaning of a λ-term by reduction/expansion w.r.t. the considered computational rules is one of the minimal requirements for a λ-model. Being the intersection type systems a general framework for the study of semantic domains for the Lambdacalculus, the present paper provides a characterisation of “meaning invariance” in terms of characterisation results for intersection type systems enabling typing invariance of terms w.r.t. various notions of reduction/expansion, like β, η and a number of relevant restrictions of theirs.

Filter Models and Easy Terms

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 filter models.

A complete characterization of complete intersection-type preorders

We characterize those type preorders which yield complete intersection-type assignment systems for λ-calculi, with respect to the three canonical set-theoretical semantics for intersection-types: the inference semantics, the simple semantics, and the F-semantics. These semantics arise by taking as interpretation of types subsets of applicative structures, as interpretation of the preorder relation, ≤, set-theoretic inclusion, as interpretation of the intersection constructor, ∩, set-theoretic intersection, and by taking the interpretation of the arrow constructor, → à la Scott, with respect to either any possible functionality set, or the largest one, or the least one.These results strengthen and generalize significantly all earlier results in the literature, to our knowledge, in at least three respects. First of all the inference semantics had not been considered before. Second, the characterizations are all given just in terms of simple closure conditions on the preorder relation, ≤, on the types, rather than on the typing judgments themselves. The task of checking the condition is made therefore considerably more tractable. Last, we do not restrict attention just to λ-models, but to arbitrary applicative structures which admit an interpretation function. Thus we allow also for the treatment of models of restricted λ-calculi. Nevertheless the characterizations we give can be tailored just to the case of λ-models.

Solutions of Functorial and Non-Functorial Metric Domain Equations

Abstract A new method for solving domain equations in categories of metric spaces is studied. The categories CMS≈ and KMS≈ are introduced, having complete and compact metric spaces as objects and ɛ-adjoint pairs as arrows. The existence and uniqueness of fixed points for certain endofunctors on these categories is established. The classes of complete and compact metric spaces are considered as pseudo-metric spaces, and it is shown how to solve domain equations in a non-categorical framework.

Simple Easy Terms

Abstract We illustrate the use of intersection types as a semantic tool for proving easiness result on λ-terms. We single out the notion of simple easiness for λ-terms as a useful semantic property for building filter models with special purpose features. Relying on the notion of easy intersection type theory, given λ-terms M and E, with E simple easy, we successfully build a filter model which equates interpretation of M and E, hence proving that simple easiness implies easiness. We finally prove that a class of λ-terms generated by ω2ω2 are simple easy, so providing alternative proof of easiness for them.

Intersection types and domain operators

We use intersection types as a tool for obtaining λ-models. Relying on the notion of easy intersection type theory, we successfully build a λ-model in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us to prove two results. The first gives a proof of consistency of the λ-theory where the λ-term (λx.xx)(λx.xx) is forced to behave as the join operator. This result has interesting consequences on the algebraic structure of the lattice of λ-theories. The second result is that for any simple easy term, there is a λ-model, where the interpretation of the term is the minimal fixed point operator.

Intersection types and lambda models

Invariance of interpretation by β-conversion is one of the minimal requirements for any standard model for the λ-calculus. With the intersection-type systems being a general framework for the study of semantic domains for the λ-calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results Ibr intersection-type assignment systems.Instead of considering conversion as a whole, reduction and expansion will be considered separately. Not only for usual computational rules like β η, but also for a number of relevant restrictions of those. Characterisations will be also provided for (intersection) filter structures that are indeed λ-models.

A fixed-point theorem in a category of compact metric spaces

Various results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces. All these results are proved for contracting functors which satisfy some further requirements, depending on the category in question. Following a new kind of approach, based on the notion of η-isometry, we show that the sole hypothesis of contractivity is enough for proving existence and uniqueness of fixed points for endofunctors on the category of compact metric spaces and embedding-projection pairs.

Partializing stone spaces using SFP domains

In this paper we investigate the problem of “partializing” Stone spaces by “Sequence of Finite Posets” (SFP) domains. More specifically, we introduce a suitable subcategory SFP m of SFP which is naturally related to the special category of Stone spaces 2-Stone by the functor MAX, which associates to each object of SFPm the space of its maximal elements. The category SFP m is closed under limits as well as many domain constructors, such as lifting, sum, product and Plotkin powerdomain. The functor MAX preserves limits and commutes with these constructors. Thus, SFP domains which “partialize” solutions of a vast class of domain equations in 2-Stone, can be obtained by solving the corresponding equations in SFP m. Furthermore, we compare two classical partializations of the space of Milners Synchronization Trees using SFP domains (see [3], [15]). Using the notion of “rigid” embedding projection pair, we show that the two domains are not isomorphic, thus providing a negative answer to an open problem raised in [15].

Tailoring Filter Models

Conditions on type preorders are provided in order to characterize the induced filter models for the λ-calculus and some of its restrictions. Besides, two examples are given of filter models in which not all the continuous functions are representable.