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Continuation semantics for the Lambek–Grishin calculus
 INFORMATION AND COMPUTATION
, 2010
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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 ..."
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Cited by 22 (21 self)
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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.
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 ..."
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Cited by 14 (2 self)
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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
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 ..."
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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
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 ..."
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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.
CurryHoward Term Calculi for GentzenStyle Classical Logic
, 2008
"... This thesis is concerned with the extension of the CurryHoward Correspondence to classical logic. Although much progress has been made in this area since the seminal paper by Griffin, we believe that the question of finding canonical calculi corresponding to classical logics has not yet been resolv ..."
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This thesis is concerned with the extension of the CurryHoward Correspondence to classical logic. Although much progress has been made in this area since the seminal paper by Griffin, we believe that the question of finding canonical calculi corresponding to classical logics has not yet been resolved. We examine computational interpretations of classical logics which we keep as close as possible to Gentzen’s original systems, equipped with general notions of reduction. We present a calculus X i which is based on classical sequent calculus and the stronglynormalising cutelimination procedure defined by Christian Urban. We examine how the notion of shallow polymorphism can be adapted to the moregeneral setting of this calculus. We show that the intuitive adaptation of these ideas fails to be sound, and give a novel solution. In the setting of classical natural deduction, we examine the lambdamu calculus of Parigot. We show that the underlying logic is incomplete in various ways, compared with a standard Gentzenstyle presentation of classical natural deduction. We relax the identified
X : a diagrammatic calculus with a classical fragrance
"... Abstract *X is a diagrammatic calculus. This means that it describes programs by 2dimensional diagrams and computations are reductions of those diagrams. In addition it has a 1dimensional syntax. Type system of *X interprets simply classical logic in a CurryHoward correspondence. Since λcalculus ..."
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Abstract *X is a diagrammatic calculus. This means that it describes programs by 2dimensional diagrams and computations are reductions of those diagrams. In addition it has a 1dimensional syntax. Type system of *X interprets simply classical logic in a CurryHoward correspondence. Since λcalculus can be easily implemented, its untyped version is Turing complete. 1
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
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 ..."
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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.