Retractions in Intersection Types

This paper deals with retraction - intended as isomorphic embedding - in intersection types building left and right inverses as terms of a lambda calculus with a bottom constant. The main result is a necessary and sufficient condition two strict intersection types must satisfy in order to assure the existence of two terms showing the first type to be a retract of the second one. Moreover, the characterisation of retraction in the standard intersection types is discussed.Comment: In Proceedings ITRS 2016, arXiv:1702.0187

The heart of intersection type assignment: Normalisation proofs revisited

AbstractThis paper gives a new proof for the approximation theorem and the characterisation of normalisability using intersection types for a system with ‘w and a ≤-relation that is contra-variant over arrow types. The technique applied is to define reduction on derivations and to show a strong normalisation result for this reduction. From this result, the characterisation of strong normalisation and the approximation result will follow easily; the latter, in its turn, will lead to the characterisation of (head) normalisability

The Heart of Intersection Type Assignment

This paper gives a new proof for the approximation theorem and the characterisation of normalisability using intersection types. The technique applied is to define reduction on derivations and to show a strong normalisation result for this reduction. From this result, the characterisation of strong normalisation and the approximation result will follow easily; the latter, in its turn, will lead to the characterisation of (head-)normalisability

The Heart of Intersection Type Assignment

This paper gives a new proof for the approximation theorem and the characterisation of normalisability using intersection types. The technique applied is to define reduction on derivations and to show a strong normalisation result for this reduction. From this result, the characterisation of strong normalisation and the approximation result will follow easily; the latter, in its turn, will lead to the characterisation of (head-)normalisability

A type system for context-dependent overloading.

This article presents a type system for context-dependent overloading, based on the notion of constrained types. These are types constrained by the definition of functions or constants of given types. This notion supports both overloading and a form of subtyping, and is related to Haskell type classes [11,2], System O [7] and other systems with constrained types. We study an extension of the Damas-Milner system [4,1] with constrained types. The inference system presented uses a context-dependent overloading policy, which is specified by means of a predicate used in a single inference rule. The idea simplifies the treatment of overloading, enables the simplification of inferred types (by means of class type annotations), and is adequate for use in a type system with higher-order types

On sets of terms having a given intersection type

Working in a variant of the intersection type assignment system of Coppo, Dezani-Ciancaglini and Venneri [1981], we prove several facts about sets of terms having a given intersection type. Our main result is that every strongly normalizing term M admits a *uniqueness typing*, which is a pair $(\Gamma,A)$ such that 1) $\Gamma \vdash M : A$ 2) $\Gamma \vdash N : A \Longrightarrow M =_{\beta\eta} N$ We also discuss several presentations of intersection type algebras, and the corresponding choices of type assignment rules. Moreover, we show that the set of closed terms with a given type is uniformly separable, and, if infinite, forms an adequate numeral system. The proof of this fact uses an internal version of the B\"ohm-out technique, adapted to terms of a given intersection type

Essential intersection type assignment

This paper introduces a notion of intersection type assignment on the Lambda Calculus that is a restriction of the BCD-system as presented in [4]. This restricted system is essential in the following sense: it is an almost syntax directed system that satisfies all major properties of the BCD-system. The set of typeable terms can be characterized in the same way, the system is complete with respect to the simple type semantics, and it has the principal type property