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Light types for polynomial time computation in lambdacalculus
 In Proceedings of the 19th IEEE Syposium on Logic in Computer Science
, 2004
"... We propose a new type system for lambdacalculus ensuring that welltyped programs can be executed in polynomial time: Dual light affine logic (DLAL). DLAL has a simple type language with a linear and an intuitionistic type arrow, and one modality. It corresponds to a fragment of Light affine logic ..."
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We propose a new type system for lambdacalculus ensuring that welltyped programs can be executed in polynomial time: Dual light affine logic (DLAL). DLAL has a simple type language with a linear and an intuitionistic type arrow, and one modality. It corresponds to a fragment of Light affine logic (LAL). We show that contrarily to LAL, DLAL ensures good properties on lambdaterms: subject reduction is satisfied and a welltyped term admits a polynomial bound on the reduction by any strategy. Finally we establish that as LAL, DLAL allows to represent all polytime functions. 1
Semantics of linear continuationpassing in callbyname
 In Proc. Functional and Logic Programming, Springer Lecture Notes in Comput. Sci
, 2004
"... Abstract. We propose a semantic framework for modelling the linear usage of continuations in typed callbyname programming languages. On the semantic side, we introduce a construction for categories of linear continuations, which gives rise to cartesian closed categories with “linear classical disj ..."
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Cited by 8 (4 self)
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Abstract. We propose a semantic framework for modelling the linear usage of continuations in typed callbyname programming languages. On the semantic side, we introduce a construction for categories of linear continuations, which gives rise to cartesian closed categories with “linear classical disjunctions ” from models of intuitionistic linear logic with sums. On the syntactic side, we give a simply typed callbyname λµcalculus in which the use of names (continuation variables) is restricted to be linear. Its semantic interpretation into a category of linear continuations then amounts to the callbyname continuationpassing style (CPS) transformation into a linear lambda calculus with sum types. We show that our calculus is sound for this CPS semantics, hence for models given by the categories of linear continuations.
A terminating and confluent linear lambda calculus
 PROC. OF 17TH INT. CONFERENCE RTA 2006, VOLUME 4098 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and ChurchRosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the eq ..."
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We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and ChurchRosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the equational theory of the linear lambda calculus. As an application we prove the strong normalization of the simply typed computational lambda calculus by giving a reductionpreserving translation into the linear lambda calculus.
Coherence of the Double Involution on *Autonomous Categories
 THEORY AND APPLICATIONS OF CATEGORY THEORY
, 2006
"... We show that any free ∗autonomous category is equivalent (in a strict sense) to a free ∗autonomous category in which the doubleinvolution (−) ∗∗ is the identity functor and the canonical isomorphism A ≃ A∗∗ is an identity arrow for all A. ..."
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We show that any free ∗autonomous category is equivalent (in a strict sense) to a free ∗autonomous category in which the doubleinvolution (−) ∗∗ is the identity functor and the canonical isomorphism A ≃ A∗∗ is an identity arrow for all A.
On categorical models of classical logic and the geometry of interaction
, 2005
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this co ..."
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this collapse by interpreting cutreduction by a posetenrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the settheoretic product. In this article, which is selfcontained, we present an improved axiomatization of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negationfree models called Dummett categories. Examples include, besides the classical categories above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from homsemilattices which have a straightforward prooftheoretic definition. Moreover, we show that the GeometryofInteraction construction can be extended from multiplicative linear logic to classical logic, by applying it to obtain a classical
COHERENCE OF THE DOUBLE INVOLUTION ON*AUTONOMOUS CATEGORIES
"... J.R.B. COCKETT, M. HASEGAWA AND R.A.G. SEELY Abstract. We show that any free *autonomous category is equivalent (in a strictsense) to a free *autonomous category in which the doubleinvolution ()* * is the identity functor and the canonical isomorphism A ' A* * is an identity arrow for all A ..."
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J.R.B. COCKETT, M. HASEGAWA AND R.A.G. SEELY Abstract. We show that any free *autonomous category is equivalent (in a strictsense) to a free *autonomous category in which the doubleinvolution ()* * is the identity functor and the canonical isomorphism A ' A* * is an identity arrow for all A.
COHERENCE OF THE DOUBLE INVOLUTION ON \LambdaAUTONOMOUS CATEGORIES
"... Fortunately, there is no such semantic gap: in this paper we provide a coherence theorem for the double involution on \Lambdaautonomous categories, which tells us that there is no difference between the uptoidentity approach and the uptoisomorphism approach, as far as this doublenegation probl ..."
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Fortunately, there is no such semantic gap: in this paper we provide a coherence theorem for the double involution on \Lambdaautonomous categories, which tells us that there is no difference between the uptoidentity approach and the uptoisomorphism approach, as far as this doublenegation problem is concerned. This remains true under the presence of linear exponential comonads and finite products (the semantic counterpart of exponentials and additives respectively). Our proof is fairly short and simple, and we suspect that this is folklore among specialists (at least everyone would expect such a result), though we are not aware of an explicit treatment of this issue in the literature.
A PATTERNMATCHING CALCULUS FOR ∗AUTONOMOUS CATEGORIES
"... ABSTRACT. This article sums up the details of a linear λcalculus that can be used as an internal language of ∗autonomous categories. The coherent isomorphisms associated with the autonomous ( = symmetric monoidal) structure are represented by pattern matching, and the inverse to the evident natura ..."
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ABSTRACT. This article sums up the details of a linear λcalculus that can be used as an internal language of ∗autonomous categories. The coherent isomorphisms associated with the autonomous ( = symmetric monoidal) structure are represented by pattern matching, and the inverse to the evident natural transformation ηA: A ✲ ((A ⊸ ⊤) ⊸ ⊤) is represented by a typeindexed family of constant CA: ((A ⊸ ⊤) ⊸ ⊤) ⊸ A. The calculus presented in this article is no great novelty. It is essentially DILL [1] minus additives, plus extended patternmatching and a typeindexed family of constants CA: ((A ⊸ ⊤) ⊸ ⊤) ⊸ A. It can also be seen as Hasegawa’s DCLL [2] minus the additive functions space →, plus pattern matching and the multiplicative conjunction ⊗. The main purpose of our calculus is the rapid verification of equalities in autonomous and ∗autonomous categories. To this end, we emphasize pattern matching, which in is very useful in practice, more strongly than in [1] or [2]. (A note to the impatient: the translation of categorical expressions into our calculus is presented in Table 5.)
MLL proof nets as Hamming Codes
, 2009
"... Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. In this paper we propose a novel approach for analyzing proof nets of Multiplicative Linear Log ..."
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Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. In this paper we propose a novel approach for analyzing proof nets of Multiplicative Linear Logic (MLL) by coding theory. We define families of proof structures and introduce a metric space for each family. In each family, 1. an MLL proof net is a real code 2. a proof structure that is not an MLL proof net is a false code. In this paper we show that in the framework one errordetecting is possible but one errorcorrecting not. Moreover, we show that affile logic and MLL + MIX are not appropriate for this framework. That explains why MLL is better than such similar logics. 1