Characterising strongly normalising intuitionistic terms

This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary lambdal-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary lambda-terms, LambdaJ-terms (lambda-terms with generalised application), and lambdax-terms (lambda-terms with explicit substitution). We explain via our system why the type systems iin the natural deduction format for LambdaJ and lambdax known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the beta-strongly normalising l-terms, as a corollary to a PSN-result, relating the lambda-calculus and the intuitionistic sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known characterisation of the beta-strongly normalising lambda-terms in terms of assignment of strict types follows as an easy corollary of our results.Fundação para a Ciência e Tecnologi

Resource control and strong normalisation

We introduce the \emph{resource control cube}, a system consisting of eight intuitionistic lambda calculi with either implicit or explicit control of resources and with either natural deduction or sequent calculus. The four calculi of the cube that correspond to natural deduction have been proposed by Kesner and Renaud and the four calculi that correspond to sequent lambda calculi are introduced in this paper. The presentation is parameterized with the set of resources (weakening or contraction), which enables a uniform treatment of the eight calculi of the cube. The simply typed resource control cube, on the one hand, expands the Curry-Howard correspondence to intuitionistic natural deduction and intuitionistic sequent logic with implicit or explicit structural rules and, on the other hand, is related to substructural logics. We propose a general intersection type system for the resource control cube calculi. Our main contribution is a characterisation of strong normalisation of reductions in this cube. First, we prove that typeability implies strong normalisation in the ''natural deduction base" of the cube by adapting the reducibility method. We then prove that typeability implies strong normalisation in the ''sequent base" of the cube by using a combination of well-orders and a suitable embedding in the ''natural deduction base". Finally, we prove that strong normalisation implies typeability in the cube using head subject expansion. All proofs are general and can be made specific to each calculus of the cube by instantiating the set of resources

Characterising strongly normalising intuitionistic sequent terms

This paper gives a characterisation, via intersection types, of the strongly normalising terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary lambda-calculus. The completeness of the typing system is obtained from subject expansion at root position. This paper's sequent term calculus integrates smoothly the lambda-terms with generalised application or explicit substitution. Strong normalisability of these terms as sequent terms characterises their typeability in certain "natural'' typing systems with intersection types. The latter are in the natural deduction format, like systems previously studied by Matthes and Lengrand et al., except that they do not contain any extra, exceptional rules for typing generalised applications or substitution

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

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