Intersection Types for the Computational lambda-Calculus

We study polymorphic type assignment systems for untyped lambda-calculi with effects, based on Moggi's monadic approach. Moving from the abstract definition of monads, we introduce a version of the call-by-value computational lambda-calculus based on Wadler's variant with unit and bind combinators, and without let. We define a notion of reduction for the calculus and prove it confluent, and also we relate our calculus to the original work by Moggi showing that his untyped metalanguage can be interpreted and simulated in our calculus. We then introduce an intersection type system inspired to Barendregt, Coppo and Dezani system for ordinary untyped lambda-calculus, establishing type invariance under conversion, and provide models of the calculus via inverse limit and filter model constructions and relate them. We prove soundness and completeness of the type system, together with subject reduction and expansion properties. Finally, we introduce a notion of convergence, which is precisely related to reduction, and characterize convergent terms via their types

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

A characterization of F-complete type assignments

AbstractThe aim of this paper is to investigate the soundness and completeness of the intersection type discipline (for terms of the (untyped λ-calculus) with respect to the F-semantics (F-soundness and F-completeness).As pointed out by Scott, if D is the domain of a γ-model, there is a subset F of D whose elements are the ‘canonical’ representatives of functions. The F-semantics of types takes into account that theintuitive meaning of “σ→τ” is ‘the type of functions with domain σ and range τ’ and interprets σ→τ as a subset of F.The type theories which induce F-complete type assignments are characterized. It follows that a type assignment is F-complete iff equal terms get equal types and, whenever M has a type ϕ∧ωn, where ϕ is a type variable and ϕ is the ‘universal’ type, the term λz1…zn…Mz1…zn has type ϕ. Here we assume that z1…z.n do not occur free in M

Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices

The lambda-calculus with de Bruijn indices assembles each alpha-class of lambda-terms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable lambda-terms through the property that a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in beta-contractions, several calculi of explicit substitution were developed mostly with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborate type systems such as intersection types. In previous work, we introduced a de Bruijn version of the lambda-calculus with an intersection type system and proved that it preserves subject reduction, a basic property of type systems. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings for beta-normal forms is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss the failure of the subject reduction property and some possible solutions for it

Characterisation of Strongly Normalising lambda-mu-Terms

We provide a characterisation of strongly normalising terms of the lambda-mu-calculus by means of a type system that uses intersection and product types. The presence of the latter and a restricted use of the type omega enable us to represent the particular notion of continuation used in the literature for the definition of semantics for the lambda-mu-calculus. This makes it possible to lift the well-known characterisation property for strongly-normalising lambda-terms - that uses intersection types - to the lambda-mu-calculus. From this result an alternative proof of strong normalisation for terms typeable in Parigot's propositional logical system follows, by means of an interpretation of that system into ours.Comment: In Proceedings ITRS 2012, arXiv:1307.784