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Linear Logic, Monads and the Lambda Calculus
 In 11 th LICS
, 1996
"... Models of intuitionistic linear logic also provide models of Moggi's computational metalanguage. We use the adjoint presentation of these models and the associated adjoint calculus to show that three translations, due mainly to Moggi, of the lambda calculus into the computational metalanguage (direc ..."
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Cited by 32 (4 self)
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Models of intuitionistic linear logic also provide models of Moggi's computational metalanguage. We use the adjoint presentation of these models and the associated adjoint calculus to show that three translations, due mainly to Moggi, of the lambda calculus into the computational metalanguage (direct, callbyname and callbyvalue) correspond exactly to three translations, due mainly to Girard, of intuitionistic logic into intuitionistic linear logic. We also consider extending these results to languages with recursion. 1. Introduction Two of the most significant developments in semantics during the last decade are Girard's linear logic [10] and Moggi's computational metalanguage [14]. Any student of these formalisms will suspect that there are significant connections between the two, despite their apparent differences. The intuitionistic fragment of linear logic (ILL) may be modelled in a linear model  a symmetric monoidal closed category with a comonad ! which satisfies some extr...
CallbyName, CallbyValue, CallbyNeed, and the Linear Lambda Calculus
, 1994
"... Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second corresponds t ..."
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Cited by 28 (5 self)
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Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second corresponds to callbyvalue. We further show that if the target of the translation is taken to be an affine calculus, where ! controls contraction but weakening is allowed everywhere, then the second translation corresponds to a callbyneed calculus, as recently defined by Ariola, Felleisen, Maraist, Odersky, and Wadler. Thus the different calling mechanisms can be explained in terms of logical translations, bringing them into the scope of the CurryHoward isomorphism.
A soft type assignment system for λcalculus
 In Proceedings of the Computer Science Logic, 21st International Workshop  CSL’07
"... Abstract. Soft Linear Logic (SLL) is a subsystem of secondorder linear logic with restricted rules for exponentials, which is correct and complete for PTIME. We design a type assignment system for the λcalculus (STA), which assigns to λterms as types (a proper subset of) SLL formulas, in such a w ..."
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Cited by 5 (2 self)
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Abstract. Soft Linear Logic (SLL) is a subsystem of secondorder linear logic with restricted rules for exponentials, which is correct and complete for PTIME. We design a type assignment system for the λcalculus (STA), which assigns to λterms as types (a proper subset of) SLL formulas, in such a way that typable terms inherit the good complexity properties of the logical system. Namely STA enjoys subject reduction and normalization, and it is correct and complete for PTIME and FPTIME.
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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Cited by 3 (2 self)
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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.
Separating Weakening and Contraction in a Linear Lambda Calculus
 in: Proc. CATS'98, Computing: the Fourth Australian Theory Symposium (Perth
, 1996
"... . We present a separatedlinear lambda calculus of resource consumption based on a refinement of linear logic which allows separate control of weakening and contraction. The calculus satisfies subject reduction and confluence, and inherits previous results on the relationship of Girard's two transla ..."
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Cited by 2 (1 self)
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. We present a separatedlinear lambda calculus of resource consumption based on a refinement of linear logic which allows separate control of weakening and contraction. The calculus satisfies subject reduction and confluence, and inherits previous results on the relationship of Girard's two translations from minimal intuitionistic logic to linear logic with callbyname and callbyvalue. We construct a hybrid translation from Girard's two which is sound and complete for mapping types and reduction sequences from callbyneed into separatedlinear . This treatment of callbyneed is more satisfying than in previous work, allowing a contrasting of all three reduction strategies in the manner (for example) that the CPS translations allow for callbyname and callbyvalue. H OW can we explain the differences between parameterpassing styles? With the continuationpassing style (CPS) transforms [24, 25], one makes the flow of control explicit. Each parameterpassing style is associated ...
Lambda! Considered Both as a Paradigmatic Language and as a MetaLanguage
"... Intuitionistic Linear Logic (ILL) is a resourceconscious logic. The CurryHoward Isomorphism (CHI) applied to ILL, generates typed functionallike languages that have primitive constants by means of which the amount of resources (terms), used during the computation, is explicit. \Gamma ! is an u ..."
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Intuitionistic Linear Logic (ILL) is a resourceconscious logic. The CurryHoward Isomorphism (CHI) applied to ILL, generates typed functionallike languages that have primitive constants by means of which the amount of resources (terms), used during the computation, is explicit. \Gamma ! is an untyped functionallike language inspired from a typed language joined at ILL by CHI. We want to use the resourceaware language \Gamma ! both as a paradigmatic programming language and as a metalanguage for implementing a fragment of the untyped calculus fi . For using \Gamma ! in the first way we give an algorithm for automatically assigning formulas of ILL as types to terms of \Gamma ! . Concerning the second kind of use, we introduce a onestep translation Tr from the fragment C of fi that can be typed a la Curry to the typable fragment of \Gamma ! in ILL. Tr preserves the linearbehaved terms of C and is both correct and complete, in a reasonable sense, w.r.t. the...
A TypeFree ResourceAware λCalculus
, 1996
"... We introduce and study a functional language λ_R , having two main features. λ_R has the same computational power of the λcalculus. λ_R enjoys the resourceawareness of the typed/typable functional languages which encode the Intuitionistic Linear Logic. ..."
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We introduce and study a functional language λ_R , having two main features. λ_R has the same computational power of the λcalculus. λ_R enjoys the resourceawareness of the typed/typable functional languages which encode the Intuitionistic Linear Logic.