Intersection Type System Research Articles

Principal Types and Unification for Simple Intersection Type Systems

In this paper we introduce a decidable restriction of the intersection type discipline. The restriction is based on a uniform principle rather than on a notion of rank. For this system we give a type checking algorithm that we prove to be sound and complete. An extended language of types with labelled intersections and intersection schemes is needed to describe the principal type of terms which turns out to be unique module a set of equations between labels. The unification algorithm far labelled types, that extends first order unification, could be used as a basis to define inference algorithms for other (more powerful) systems with intersection.

Embedding a Second-Order Type System into an Intersection Type System

This paper presents the relationship between a second-order type assignment system T∀ and an intersection type assignment system T∧. First we define a translation tr from intersection types to second-order types. Then we define a system T∧* obtained from T∧ by restricting the use of the intersection type introduction rule, and show that T∧* and T∀ are equivalent in the following senses: (a) if a λ-term M has a type σ in T∧*, then M has the type tr(σ) in T∀; and conversely, (b) if M has a type T in T∀, then M has a type σ in T∧* such that tr(σ) is equivalent to T. These two theorems mean that T∀ is embedded into T∧.

Type inference, abstract interpretation and strictness analysis

Filter domains (Coppo et al.,1984) can be seen as abstract domains for the interpretation of (functional) type-free programming languages. What is remarkable is the fact that in filter domains the interpretation of a term is given by the set of its types in the intersection type discipline with inclusion, thus reducing the computation of an abstract interpretation to typechecking. As a main example, an abstract filter domain for strictness analysis of type-free functional languages is presented. The inclusion relation between types representing strictness properties has a complete recursive axiomatization. Type inference rules cannot be complete (strictness being a Π 0 1 property), but a complete extension of the type inference system is presented.

Principal Type Schemes for the Strict Type Assignment System

We study the strict type assignment system, a restriction of the intersection type discipline, and prove that it has the principal type property. We define, for a term M, the principal pair (of basis and type). We specify three operations on pairs, and prove that all pairs deducible for M can be obtained from the principal one by these operations, and that these map deducible pairs to deducible pairs.

Complete restrictions of the intersection type discipline

In this paper the intersection type discipline as defined in Barendregt (1983) is studied. We will present two different and independent complete restrictions of the intersection type discipline. The first restricted system, the strict type assignment system, is presented in Section 2. Its major feature is the absence of the derivation rule (⩽) and it is based on a set of strict types. We will show that these together give rise to a strict filter lambda model that is essentially different from the one presented in Barendregt. We will show that the strict type assignment system is the nucleus of the full system, i.e. for every derivation in the intersection type discipline there is a derivation in which (⩽) is used only at the very end. Finally we will prove that strict type assignment is complete for inference semantics. The second restricted system is presented in Section 3. Its major feature is the absence of the type ω. We will show that this system gives rise to a filter λ I-model and that type assignment without ω is complete for the λ I-calculus. Finally we will prove that a lambda term is typeable in this system if and only if it is strongly normalizable.

Principal type scheme and unification for intersection type discipline

The intersection type discipline for the λ-calculus (ITD) is an extension of the classical functionality theory of Curry. In the ITD a term satisfying a given property has a principal type scheme in an extended meaning, i.e., there is a type scheme deducible for it from which all and only the type schemes deducible for it are reachable, by means of suitable operations. The problem of finding the principal type scheme for a term, if it exists, is semidecidable. In the paper a procedure is shown, building the principal type scheme of a term through the construction of the most general unifier for intersection type schemes.