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Categorical Models for Intuitionistic and Linear Type Theory
 In Foundations of Software Science and Computation Structure (FoSSaCS 2000), Springer Lecture Notes in Comput. Sci. 1784
, 2000
"... This paper describes the categorical semantics of a system of mixed intuitionistic and linear type theory (ILT). ILT was proposed by G. Plotkin and also independently by P. Wadler. The logic associated with ILT is obtained as a combination of intuitionistic logic with intuitionistic linear logic, an ..."
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This paper describes the categorical semantics of a system of mixed intuitionistic and linear type theory (ILT). ILT was proposed by G. Plotkin and also independently by P. Wadler. The logic associated with ILT is obtained as a combination of intuitionistic logic with intuitionistic linear logic, and can be embedded in Barber and Plotkin's Dual Intuitionistic Linear Logic (DILL). However, unlike DILL, the logic for ILT lacks an explicit modality ! that translates intuitionistic proofs into linear ones. So while the semantics of DILL can be given in terms of monoidal adjunctions between symmetric monoidal closed categories and cartesian closed categories, the semantics of ILT is better presented via fibrations. These interpret double contexts, which cannot be reduced to linear ones. In order to interpret the intuitionistic and linear identity axioms acting on the same type we need fibrations satisfying the comprehension axiom.
Resource operators for λcalculus
 INFORM. AND COMPUT
, 2007
"... We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties ..."
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We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simplytyped terms, step by step simulation of βreduction and full composition.
Linear explicit substitutions (extended abstract
 In Proceedings of WESTAPP'98
, 1998
"... Thecalculus [1] adds explicit substitutions to thecalculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises thecalculus to provide a linear calculus of explicit substitutions which analogously descr ..."
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Thecalculus [1] adds explicit substitutions to thecalculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises thecalculus to provide a linear calculus of explicit substitutions which analogously describes the implementation of linear functional programming languages. 1
An Efficient Linear Machine With SinglePointer Property
"... We introduce xLIN, a novel linear abstract machine that introduces no penalty in computing with nonlinear resources, while ensuring that computing with linear resources takes no space and constant time. Garbage collection is not required for linear objects, while still present for nonlinear object ..."
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We introduce xLIN, a novel linear abstract machine that introduces no penalty in computing with nonlinear resources, while ensuring that computing with linear resources takes no space and constant time. Garbage collection is not required for linear objects, while still present for nonlinear objects. We propose a weak calculus of explicit substitutions as its foundations, and show how it may be easily `derived' from it. We illustrate in which way our machine may be seen as a refinement of Krivine's abstract machine, and hint at how nonlinear computation could be optimized to serve as framework for the implementation of functional languages. 1 Introduction Since linear logic first appeared [7], it was immediately perceived by the scientific community as a promising framework in which the mechanics of reduction could be better understood. By virtue of the existence of a strong connection between the world of logic and that of the typed calculus (by the so called CurryHoward isomorph...
Explicit Substitutions for Linear Logical
 Proceedings of the Worskshop on Logical Frameworks and MetaLanguages — LFM’99
, 1999
"... We present the calculus xdLLF and experiment with aspects of its metatheory. xdLLF integrates linear explicit substitutions in de Bruijn notation into the simplytyped fragment of the linear logical framework LLF. After observing that the expected rules invalidate subject reduction, we devise a sp ..."
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We present the calculus xdLLF and experiment with aspects of its metatheory. xdLLF integrates linear explicit substitutions in de Bruijn notation into the simplytyped fragment of the linear logical framework LLF. After observing that the expected rules invalidate subject reduction, we devise a specication of normalization inspired by the bigstep semantics of programming languages, and prove it correct. 1 Introduction Explicit substitutions [1] have been used to rationalize the implementation of many systems based on various calculi, such as functional languages, logical frameworks, and higherorder logic programming languages. As linear calculi have grown in popularity, so has the need for solid and ecient support for their implementation. A linear adaptation of explicit substitution techniques is a prime candidate. The authors of this paper have separately explored this possibility in two distinct settings: In [6], Ghani, de Paiva, and Ritter have designed the language...
An Efficient Linear Machine With UpdateinPlace for Linear Variables
"... We introduce xLIN, a novel linear abstract machine that ensures computing with linear resources takes no space and constant time by executing linear substitutions immediately, as specied by the rule. In addition, computing with nonlinear resources introduces no e ciency penalty. Garbage colle ..."
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We introduce xLIN, a novel linear abstract machine that ensures computing with linear resources takes no space and constant time by executing linear substitutions immediately, as specied by the rule. In addition, computing with nonlinear resources introduces no e ciency penalty. Garbage collection is not required for linear objects, while still present for nonlinear objects. Using a suitable memory model we show that when we use sharing of common subexpression during execution the immediate substitution cannot be done for all linear variables and identify via a type system which incorporates also storage locations a subset of linear variables which admits immediate substitution. This subset is rich enough to include many standard recursively dened linear functions. 1 Introduction Since linear logic rst appeared [7], it was immediately perceived by the scientic community as a promising framework in which the mechanics of reduction could be better understood. By vir...