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Classical linear logic of implications
 In Proc. Computer Science Logic (CSL'02), Springer Lecture Notes in Comp. Sci. 2471
, 2002
"... Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technica ..."
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Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technically managable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound andcomplete for categorytheoretic models given by ∗autonomous categories with linear exponential comonads. 1
A classical linear lambdacalculus
, 1996
"... This paper proposes and studies a typed calculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural ..."
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Cited by 9 (0 self)
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This paper proposes and studies a typed calculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. This formulation is compared in detail to the sequent calculus formulation. In an appendix I shall also demonstrate a somewhat hidden connexion with the paradigm of control operators for functional languages which gives a new computational interpretation of Parigot's techniques.
Linearly Used Continuations
 Computer Science Department, Indiana University
, 2001
"... this paper is to describe the main conceptual aspects of linearly used continuations in a way that keeps the technical discussion as simple as possible. So we concentrate on soundness only. A comprehensive analysis of completeness properties of our transforms, or variants, represents a challenge for ..."
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Cited by 8 (2 self)
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this paper is to describe the main conceptual aspects of linearly used continuations in a way that keeps the technical discussion as simple as possible. So we concentrate on soundness only. A comprehensive analysis of completeness properties of our transforms, or variants, represents a challenge for future work, and in stating the transforms for a variety of features we hope to make clear what some of the challenges are. Several of these problems are discussed at the end of the paper
Linear Logic, Comonads and Optimal Reductions
 Fundamentae Informaticae
, 1993
"... The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the calculus. In particular, we relate the two "brackets" in [GAL92a] to the two operations associated with the comonad "!" of Linear Logic. The rewriting rules can be ..."
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Cited by 7 (3 self)
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The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the calculus. In particular, we relate the two "brackets" in [GAL92a] to the two operations associated with the comonad "!" of Linear Logic. The rewriting rules can be then understood as a "local implementation" of naturality laws, that is as the broadcasting of some information from the output to the inputs of a term, following its connected structure. 1 Introduction More than fifteen years ago, L'evy [Le78] proposed a theoretical notion of optimality for calculus normalization. Roughly speaking, a reduction technique is optimal if it is able to profit of all the sharing expressed in initial term, avoiding useless duplications. For a long time, no implementation was able to achieve L'evy's performance (see [Fie90] for a quick survey). People started already to doubt of the existence of optimal evaluators, when Lamping and Kathail independently found a solution [Lam90,Ka90]...
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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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.
Sharing Continuations: Proofnets for Languages With Explicit Control
"... sumes it. Yet evaluating expressions is very familiar, while evaluating continuations is considered esoteric, even though both are made ofthe same stuff. The incorporation ofcontinuations as firstclass citizens in programming languages was not welcomed like the Emancipation Proclamation, but instea ..."
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sumes it. Yet evaluating expressions is very familiar, while evaluating continuations is considered esoteric, even though both are made ofthe same stuff. The incorporation ofcontinuations as firstclass citizens in programming languages was not welcomed like the Emancipation Proclamation, but instead regarded warily as a kind ofwitchcraft, with implementation pragmatics that are illdefined and unclear. Ifexpressions and continuations are indeed dual, then so should be the technology oftheir implementation, and the flexibility with which we reason about them. Efficient evaluation ofone should reveal dual strategies for evaluating the other. In short, everything we know about expressions we ought to know about continuations. We take a significant step towards this equality by formulating a general version ofgraph reduction that implements the sharing and optimal incremental evaluation ofboth expressions and continuations, each evaluated using the same primitive operations. By founding our technology on generic tools from logic and programming language theory, specifically the CPS transform and its relation
Linear ContinuationPassing
 IN THE 2001 ACM SIGPLAN WORKSHOP ON CONTINUATIONS (CW'01
, 2002
"... Continuations can be used to explain a wide variety of control behaviours, including calling/returning (procedures), raising/handling (exceptions), labelled jumping (goto statements), process switching (coroutines), and backtracking. However, continuations are often manipulated in a highly stylised ..."
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Continuations can be used to explain a wide variety of control behaviours, including calling/returning (procedures), raising/handling (exceptions), labelled jumping (goto statements), process switching (coroutines), and backtracking. However, continuations are often manipulated in a highly stylised way, and we show that all of these, bar backtracking, in fact use their continuations linearly ; this is formalised by taking a target language for cps transforms that has both intuitionistic and linear function types.
Acceptors as values: Functional programming in classical linear logic
, 1991
"... Girard’s linear logic has been previously applied to functional programming for performing statemanipulation and controlling storage reuse. These applications only use intuitionistic linear logic, the subset of linear logic that embeds intuitionistic logic. Full linear logic (called classical linea ..."
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Girard’s linear logic has been previously applied to functional programming for performing statemanipulation and controlling storage reuse. These applications only use intuitionistic linear logic, the subset of linear logic that embeds intuitionistic logic. Full linear logic (called classical linear logic) is much richer than this subset. In this paper, we consider the application of classical linear logic to functional programming. The negative types of linear logic are interpreted as denoting acceptors. An acceptor is an entity which takes an input of some type and returns no output. Acceptors generalize continuations and also single assignment variables, as found in data flow languages and logic programming languages. The parallel disjunction operator allows such acceptors to be used in a nontrivial fashion. Finally, the “why not ” operator of linear logic gives rise to nondeterministic values. We define a typed functional language based on the these ideas and demonstrate its use via examples. The language has a reduction semantics that generalizes typed lambda calculus, and satisfies strong normalization and ChurchRosser properties.
Equational Axiomatization of CallbyName Delimited Control
"... Control operators for delimited continuations are useful in various fields such as partial evaluation, CPS translation, and representation of monadic effects. While many works in the literature study them in callbyvalue, several recent works have shown callbyname delimited control operators are ..."
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Control operators for delimited continuations are useful in various fields such as partial evaluation, CPS translation, and representation of monadic effects. While many works in the literature study them in callbyvalue, several recent works have shown callbyname delimited control operators are also worth studying. In this paper, we study semantic foundation of the callbyname variant of the delimitedcontrol operators “shift ” and “reset”. In particular, we give a set of directstyle equations as axioms for them, and prove that it is sound and complete with respect to the CPS translation by Biernacka and Biernacki. The key observations in our proof are (1) we need to use the linearity of certain variables in the CPS terms, and (2) we must distinguish continuation variables from ordinary variables in the source terms. We also show that our axiomatization holds for the typed calculus.
Towards a Classical Linear λcalculus
 PROC. OF THE TOKYO CONFERENCE ON LINEAR LOGIC
, 1996
"... This paper considers a typed calculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction ..."
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This paper considers a typed calculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. I shall also demonstrate a somewhat hidden connexion with the continuationpassing paradigm which gives a new computational interpretation of Parigot's techniques and possibly a new style of continuation programming.