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 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

Resource control and strong normalisation (old version)

We introduce the 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 paramatrized 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

LIPIcs, Volume 97, TYPES\u2716, Complete Volume

LIPIcs, Volume 97, TYPES\u2716, Complete Volum

Front Matter, Table of Contents, Preface, Conference Organization

Front Matter, Table of Contents, Preface, Conference Organizatio

Intersection types fit well with resource control

International audienc

Intuitionistic Sequent-Style Calculus with Explicit Structural Rules

International audienceIn this paper we extend the Curry-Howard correspondence to intuitionistic sequent calculus with explicit weakening and contraction. We study a system derived from /\-Gtz of Espirito Santo by adding explicit operators for weakening and contraction, which we call l/\-Gtz. This system contains only linear terms. For the proposed calculus we introduce the type assignment system with simple types. The presented system has a natural diagrammatic representation, which is used for proving the subject reduction property. We prove the strong normalisation property by embedding l/\-Gtz into the simply typed /\lxr calculus of Kesner and Lengrand