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

A Theory of Explicit Substitutions with Safe and Full Composition

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference "International Colloquium on Automata, Languages and Programming 2008" edited by Giuseppe Castagna and Igor Walukiewic

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

Resource control and intersection types: an intrinsic connection

In this paper we investigate the $\lambda$ -calculus, a $\lambda$-calculus enriched with resource control. Explicit control of resources is enabled by the presence of erasure and duplication operators, which correspond to thinning and con-traction rules in the type assignment system. We introduce directly the class of $\lambda$ -terms and we provide a new treatment of substitution by its decompo-sition into atomic steps. We propose an intersection type assignment system for $\lambda$ -calculus which makes a clear correspondence between three roles of variables and three kinds of intersection types. Finally, we provide the characterisation of strong normalisation in $\lambda$ -calculus by means of an in-tersection type assignment system. This process uses typeability of normal forms, redex subject expansion and reducibility method.Comment: arXiv admin note: substantial text overlap with arXiv:1306.228

Quantitative Types for Intuitionistic Calculi

We define quantitative type systems for two intuitionistic term languages. While the first language in natural deduction style is already known in the literature, the second one is one of the contributions of the paper, and turns out to be a natural computational interpretation of sequent calculus style by means of a non-idempotent type discipline. The type systems are able to characterize linear-head, weak and strong normalization sets of terms. All such characterizations are given by means of combinatorial arguments, i.e. there is a measure based on type derivations which is decreasing with respect to the different reduction relations considered in the paper

Intersection types for explicit substitutions

We present a new system of intersection types for a composition-free calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical intersection types system, which characterized head normalization and weak normalization, but was not complete for strong normalization. An important role is played by the notion of available variable in a term, which is a generalization of the classical notion of free variable