Results 1  10
of
19
Computation with classical sequents
 MATHEMATICAL STRUCTURES OF COMPUTER SCIENCE
, 2008
"... X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X ..."
Abstract

Cited by 16 (16 self)
 Add to MetaCart
X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X, we will show how elaborate calculi can be embedded, like the λcalculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.
Continuation semantics for the Lambek–Grishin calculus
 INFORMATION AND COMPUTATION
, 2010
"... ..."
Approaches to polymorphism in classical sequent calculus
 In ESOP’06, LNCS 3924
, 2006
"... Abstract. X is a relatively new calculus, invented to give a CurryHoward correspondence with Classical Implicative Sequent Calculus. It is already known to provide a very expressive language; embeddings have been defined of the λcalculus, Bloo and Rose’s λx, Parigot’s λµ and Curien and Herbelin’s λ ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
Abstract. X is a relatively new calculus, invented to give a CurryHoward correspondence with Classical Implicative Sequent Calculus. It is already known to provide a very expressive language; embeddings have been defined of the λcalculus, Bloo and Rose’s λx, Parigot’s λµ and Curien and Herbelin’s λµ˜µ. We investigate various notions of polymorphism in the context of the Xcalculus. In particular, we examine the first class polymorphism of System F, and the shallow polymorphism of ML. We define analogous systems based on the Xcalculus, and show that these are suitable for embedding the original calculi. In the case of shallow polymorphism we obtain a more general calculus than ML, while retaining its useful properties. A typeassignment algorithm is defined for this system, which generalises Milner’s W. 1
Symmetric categorial grammar
 Journal of Philosophical Logic
, 2009
"... is lost or not), is a phenomenon which a linguistic semantics ought to explain, rather than ignore. (van Benthem 1986, p 213) The LambekGrishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of typeforming operations (product and residual left a ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
is lost or not), is a phenomenon which a linguistic semantics ought to explain, rather than ignore. (van Benthem 1986, p 213) The LambekGrishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of typeforming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the CurryHoward derivational semantics, and structurepreservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural language form and meaning. 1 1
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ ˜µ
 Annals of Pure and Applied Logic
"... This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minima ..."
Abstract

Cited by 6 (6 self)
 Add to MetaCart
This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minimal requirements are for a system for λµ ˜µ to be closed under subject expansion; this coincides with System M ∩ ∪ , the notion defined in [19]; however, we show that this system is not closed under subject reduction, so our goal cannot be achieved. We will then show that System M ∩ ∪ is also not closed under subjectexpansion, but can recover from this by presenting System M C as an extension of M ∩ ∪ (by adding typing rules) and showing that it satisfies subject expansion; it still lacks subject reduction. We show how to restrict M ∩ ∪ so that it satisfies subjectreduction as well by limiting the applicability to type assignment rules, but only when limiting reduction to (confluent) callbyname or callbyvalue reduction M ∩ ∪ ; in restricting the system, we sacrifice subject expansion. These results combined show that a sound and complete intersection and union type assignment system cannot be defined for λµ ˜µ with respect to full reduction.
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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.
Strong Normalisation of CutElimination that Simulates βReduction
"... This paper is concerned with strong normalisation of cutelimination for a standard intuitionistic sequent calculus. The cutelimination procedure is based on a rewrite system for proofterms with cutpermutation rules allowing the simulation of βreduction. Strong normalisation of the typed terms i ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
This paper is concerned with strong normalisation of cutelimination for a standard intuitionistic sequent calculus. The cutelimination procedure is based on a rewrite system for proofterms with cutpermutation rules allowing the simulation of βreduction. Strong normalisation of the typed terms is inferred from that of the simplytyped λcalculus, using the notions of safe and minimal reductions as well as a simulation in NederpeltKlop’s λIcalculus. It is also shown that the typefree terms enjoy the preservation of strong normalisation (PSN) property with respect to βreduction in an isomorphic image of the typefree λcalculus.
Reduction in X does not agree with Intersection and Union Types
, 2008
"... This paper defines intersection and union type assignment for the calculus X, a substitution free language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is the minimal one closed for subjectexpansion, and show th ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
This paper defines intersection and union type assignment for the calculus X, a substitution free language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is the minimal one closed for subjectexpansion, and show that it needs to be restricted to satisfy subjectreduction as well, making it unsuitable to define a semantics.