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Higherorder pattern complement and the strict λcalculus
 ACM Trans. Comput. Logic
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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.
Categorical models for explicit substitutions
 Proc. of FoSSaCS'99
, 1999
"... This paper concerns itself with the categorical semantics ofcalculi extended with explicit substitutions. For the simplytypedcalculus, indexed categories seem to provide the right categorical framework but because these structures are inherently nonlinear, alternate models are needed for linear ..."
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This paper concerns itself with the categorical semantics ofcalculi extended with explicit substitutions. For the simplytypedcalculus, indexed categories seem to provide the right categorical framework but because these structures are inherently nonlinear, alternate models are needed for linearcalculi extended with explicit substitutions. We propose to replace indexed categories by presheaves and obtain a semantics which can be specialised to both the linear and the intuitionistic case. The basic models of a calculi of linear (or intuitionistic) explicit substitutions are called linear (or cartesian) contexthandling categories. Then we add extra categorical structure to model the connectives of the logic, obtaining Lcategories as models of the (;I;,)fragment ofintuitionistic linear logic and Ecategories as models of the simply typedcalculus. The!type constructor is then modelled by a monoidal adjunction between an E and Lcategory. Finally, soundness and completeness of our categorical model is proven. 1
Linear Explicit Substitutions (Extended Abstract)
"... Abstract The *oecalculus [1] adds explicit substitutions to the *calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the *oecalculus to provide a linear calculus of explicit substitutions whic ..."
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Abstract The *oecalculus [1] adds explicit substitutions to the *calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the *oecalculus to provide a linear calculus of explicit substitutions which analogously describes the implementation of linear functional programming languages. 1 Introduction Why explicit substitutions? Traditional functional programming languages are based on the *calculus and model computation by fireduction. In its usual form (*x:t)u)t[u=x] the argument u is substituted into the body of the *abstraction in one step. This is highly inefficient from the perspective of implementations as each redex in u may be copied arbitrarily often. Abstract reduction machines try avoid this problem by reducing terms in an environment the contraction of a firedex creates a new substitution which, instead of being applied to the body of the *abstraction, is added to the existing environment and only evaluated when actually needed.
Chapter 1 An Abstract Machine Based on Linear Logic and Explicit Substitutions
"... Abstract. We present xLIN, a novel linear abstract machine that introduces no penalty in computing with nonlinear resources, while computing with linear resources might, in principle, be as fast as a series of destructive updates. Garbage collection would not be required for linear objects, while s ..."
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Abstract. We present xLIN, a novel linear abstract machine that introduces no penalty in computing with nonlinear resources, while computing with linear resources might, in principle, be as fast as a series of destructive updates. Garbage collection would not be required for linear objects, while still present for nonlinear objects. We propose a weak calculus of explicit substitutions as its foundations, and show howitmay be easily \derived &quot; from it. A proof of correctness, then, comes for free. We illustrate in which way this machine may beseen as a re nement of Krivine's abstract machine, and hint athow nonlinear computation could be optimized to serve as framework for the implementation of functional languages. Besides, not to sound so optimistic at this rather preliminary stage of development, we comment on the inherent incompatibilities that exist between the techniques suggested by the highlevel description of the machine. 1
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...