Inhabitation for Non-idempotent Intersection Types

The inhabitation problem for intersection types in the lambda-calculus is known to be undecidable. We study the problem in the case of non-idempotent intersection, considering several type assignment systems, which characterize the solvable or the strongly normalizing lambda-terms. We prove the decidability of the inhabitation problem for all the systems considered, by providing sound and complete inhabitation algorithms for them

Polymorphic intersection type assignment for rewrite systems with abstraction and -rul

We define two type assignment systems for first-order rewriting extended with application, -abstraction, and -reduction, using a combination of intersection types and second-order polymorphic types. The first system is the general one, for which we prove subject reduction, and strong normalisation of typeable terms. The second is a decidable subsystem of the first, by restricting to rank 2 (intersection and quantified) types. For this system we define, using an extended notion of unification, a notion of principal typing which is more general than ML’s principal type property, since also the types for the free variables of terms are inferre

Semantic Types for Class-based Objects

We investigate semantics-based type assignment for class-based object-oriented programming. Our motivation is developing a theoretical basis for practical, expressive, type-based analysis of the functional behaviour of object-oriented programs. We focus our research using Featherweight Java, studying two notions of type assignment:- one using intersection types, the other a ‘logical’ restriction of recursive types. We extend to the object-oriented setting some existing results for intersection type systems. In doing so, we contribute to the study of denotational semantics for object-oriented languages. We define a model for Featherweight Java based on approximation, which we relate to our intersection type system via an Approximation Result, proved using a notion of reduction on typing derivations that we show to be strongly normalising. We consider restrictions of our system for which type assignment is decidable, observing that the implicit recursion present in the class mechanism is a limiting factor in making practical use of the expressive power of intersection types. To overcome this, we consider type assignment based on recursive types. Such types traditionally suffer from the inability to characterise convergence, a key element of our approach. To obtain a semantic system of recursive types for Featherweight Java we study Nakano’s systems, whose key feature is an approximation modality which leads to a ‘logical’ system expressing both functional behaviour and convergence. For Nakano’s system, we consider the open problem of type inference. We introduce insertion variables (similar to the expansion variables of Kfoury and Wells), which allow to infer when the approximation modality is required. We define a type inference procedure, and conjecture its soundness based on a technique of Cardone and Coppo. Finally, we consider how Nakano’s approach may be applied to Featherweight Java and discuss how intersection and logical recursive types may be brought together into a single system

A journey through resource control lambda calculi and explicit substitution using intersection types (an account)

In this paper we invite the reader to a journey through three lambda calculi with resource control: the lambda calculus, the sequent lambda calculus, and the lambda calculus with explicit substitution. All three calculi enable explicit control of resources due to the presence of weakening and contraction operators. Along this journey, we propose intersection type assignment systems for all three resource control calculi. We recognise the need for three kinds of variables all requiring different kinds of intersection types. Our main contribution is the characterisation of strong normalisation of reductions in all three calculi, using the techniques of reducibility, head subject expansion, a combination of well-orders and suitable embeddings of terms

The Delta-calculus: syntax and types

International audienceWe present the $∆$-calculus, an explicitly typed $λ$-calculus with strong pairs, projections and explicit type coercions. The calculus can be parametrized with different intersection type theories T , e.g. the Coppo-Dezani, the Coppo-Dezani-Salle', the Coppo-Dezani-Venneri and the Barendregt-Coppo-Dezani ones, producing a family of $∆$-calculi with related intersection typed systems. We prove the main properties like Church-Rosser, unicity of type, subject reduction, strong normalization, decidability of type checking and type reconstruction. We state the relationship between the intersection type assignment systems a` la Curry and the corresponding intersection typed systems a` la Church by means of an essence function translating an explicitly typed $∆$-term into a pure $λ$-term one. We finally translate a $∆$-term with type coercions into an equivalent one without them; the translation is proved to be coherent because its essence is the identity. The generic $∆$- calculus can be parametrized to take into account other intersection type theories as the ones in the Barendregt et al. book

Functional Type Assignment for Featherweight Java

Abstract. We consider functional type assignment for the class-based objectoriented calculus Featherweight Java. We start with an intersection type assignment systems for this calculus for which types are preserved under conversion. We then define a variant for which type assignment is decidable, and define a notion of unification as well as a principal typeing algorithm. We show the expressivity of both our calculus and our type system by defining an encoding of Combinatory Logic into our calculus and showing that this encoding preserves typeability. We thus demonstrate that the great capabilities of functional types can be applied to the context of class-based object orientated programming