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ContinuationPassing Style and Strong Normalisation for Intuitionistic Sequent Calculi
"... Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµ˜µcalculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea co ..."
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Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµ˜µcalculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea coming from Ikeda and Nakazawa’s translation of Parigot’s λµcalculus. The embedding simulates reductions while usual continuationpassingstyle transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need “units of garbage ” to be passed. We apply the same method to other calculi, namely successive extensions of the simplytyped λcalculus leading to our intuitionistic system, and already for the simplest extension we consider (λcalculus with generalised application), this yields the first proof of strong normalisation through a reductionpreserving embedding. 1
Monadic translation of intuitionistic sequent calculus
, 2009
"... This paper proposes and analyses a monadic translation of an intuitionistic sequent calculus. The source of the translation is a typed λcalculus previously introduced by the authors, corresponding to the intuitionistic fragment of the callbyname variant of λµ˜µ of Curien and Herbelin, and the tar ..."
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This paper proposes and analyses a monadic translation of an intuitionistic sequent calculus. The source of the translation is a typed λcalculus previously introduced by the authors, corresponding to the intuitionistic fragment of the callbyname variant of λµ˜µ of Curien and Herbelin, and the target is a variant of Moggi’s monadic metalanguage, where the rewrite relation includes extra permutation rules that may be seen as variations of the “associativity ” of bind (the Kleisli extension operation of the monad). The main result is that the monadic translation simulates reduction strictly, so that strong normalisation (which is enjoyed at the target, as we show) can be lifted from the target to the source. A variant translation, obtained by adding an extra monad application in the translation of types, still enjoys strict simulation, while requiring one fewer extra permutation rule from the target. Finally we instantiate, for the cases of the identity monad and the continuations monad, the metalanguage into the simplytyped λcalculus. By this means, we give a generic account of translations of sequent calculus into natural deduction, which encompasses the traditional mapping studied by Zucker and Pottinger, and CPS translations of intuitionistic sequent calculus.
Characterising strongly normalising intuitionistic sequent terms
"... Abstract. 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 λcalc ..."
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Abstract. 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 λcalculus. The completeness of the typing system is obtained from subject expansion at root position. This paper’s sequent term calculus integrates smoothly the λterms 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 strong normalisation
, 2012
"... 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 Kesn ..."
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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 CurryHoward 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 wellorders 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