Liquid Intersection Types

We present a new type system combining refinement types and the expressiveness of intersection type discipline. The use of such features makes it possible to derive more precise types than in the original refinement system. We have been able to prove several interesting properties for our system (including subject reduction) and developed an inference algorithm, which we proved to be sound.Comment: In Proceedings ITRS 2014, arXiv:1503.0437

Intersection types for unbind and rebind

We define a type system with intersection types for an extension of lambda-calculus with unbind and rebind operators. In this calculus, a term with free variables, representing open code, can be packed into an "unbound" term, and passed around as a value. In order to execute inside code, an unbound term should be explicitly rebound at the point where it is used. Unbinding and rebinding are hierarchical, that is, the term can contain arbitrarily nested unbound terms, whose inside code can only be executed after a sequence of rebinds has been applied. Correspondingly, types are decorated with levels, and a term has type decorated with k if it needs k rebinds in order to reduce to a value. With intersection types we model the fact that a term can be used differently in contexts providing different numbers of unbinds. In particular, top-level terms, that is, terms not requiring unbinds to reduce to values, should have a value type, that is, an intersection type where at least one element has level 0. With the proposed intersection type system we get soundness under the call-by-value strategy, an issue which was not resolved by previous type systems.Comment: In Proceedings ITRS 2010, arXiv:1101.410

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

A Finite Model Property for Intersection Types

We show that the relational theory of intersection types known as BCD has the finite model property; that is, BCD is complete for its finite models. Our proof uses rewriting techniques which have as an immediate by-product the polynomial time decidability of the preorder <= (although this also follows from the so called beta soundness of BCD).Comment: In Proceedings ITRS 2014, arXiv:1503.0437

Bounding normalization time through intersection types

Non-idempotent intersection types are used in order to give a bound of the length of the normalization beta-reduction sequence of a lambda term: namely, the bound is expressed as a function of the size of the term.Comment: In Proceedings ITRS 2012, arXiv:1307.784

Mixin Composition Synthesis based on Intersection Types

We present a method for synthesizing compositions of mixins using type inhabitation in intersection types. First, recursively defined classes and mixins, which are functions over classes, are expressed as terms in a lambda calculus with records. Intersection types with records and record-merge are used to assign meaningful types to these terms without resorting to recursive types. Second, typed terms are translated to a repository of typed combinators. We show a relation between record types with record-merge and intersection types with constructors. This relation is used to prove soundness and partial completeness of the translation with respect to mixin composition synthesis. Furthermore, we demonstrate how a translated repository and goal type can be used as input to an existing framework for composition synthesis in bounded combinatory logic via type inhabitation. The computed result is a class typed by the goal type and generated by a mixin composition applied to an existing class

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

On Isomorphism of "Functional" Intersection and Union Types

Type isomorphism is useful for retrieving library components, since a function in a library can have a type different from, but isomorphic to, the one expected by the user. Moreover type isomorphism gives for free the coercion required to include the function in the user program with the right type. The present paper faces the problem of type isomorphism in a system with intersection and union types. In the presence of intersection and union, isomorphism is not a congruence and cannot be characterised in an equational way. A characterisation can still be given, quite complicated by the interference between functional and non functional types. This drawback is faced in the paper by interpreting each atomic type as the set of functions mapping any argument into the interpretation of the type itself. This choice has been suggested by the initial projection of Scott's inverse limit lambda-model. The main result of this paper is a condition assuring type isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437