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Games and Full Completeness for Multiplicative Linear Logic
 JOURNAL OF SYMBOLIC LOGIC
, 1994
"... We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the den ..."
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Cited by 210 (26 self)
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We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cutfree proof net. A key role is played by the notion of historyfree strategy; strong connections are made between historyfree strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass et al.
A New Deconstructive Logic: Linear Logic
, 1995
"... The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different a ..."
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Cited by 102 (11 self)
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The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...
The Geometry of Optimal Lambda Reduction
, 1992
"... Lamping discovered an optimal graphreduction implementation of the calculus. Simultaneously, Girard invented the geometry of interaction, a mathematical foundation for operational semantics. In this paper, we connect and explain the geometry of interaction and Lamping's graphs. The geometry of int ..."
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Cited by 97 (2 self)
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Lamping discovered an optimal graphreduction implementation of the calculus. Simultaneously, Girard invented the geometry of interaction, a mathematical foundation for operational semantics. In this paper, we connect and explain the geometry of interaction and Lamping's graphs. The geometry of interaction provides a suitable semantic basis for explaining and improving Lamping's system. On the other hand, graphs similar to Lamping's provide a concrete representation of the geometry of interaction. Together, they offer a new understanding of computation, as well as ideas for efficient and correct implementations.
A CurryHoward foundation for functional computation with control
 In Proceedings of ACM SIGPLANSIGACT Symposium on Principle of Programming Languages
, 1997
"... We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatib ..."
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Cited by 77 (3 self)
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We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant callbyvalue programming language ¯pcf v is obtained from ¯ v by augmenting it by basic arithmetic, conditionals and fixpoints. We study the behavioural properties of ¯pcf v and show that, though simple, it is a very general language for functional computation with control: it can express all the main control constructs such as exceptions and firstclass continuations. Prooftheoretically the dual ¯ v constructs of naming and ¯abstraction witness the introduction and elimination rules of absurdity respectively. Computationally they give succinct expression to a kind of generic (forward) "jump" operator, which may be regarded as a unif...
Hypercoherences: A Strongly Stable Model of Linear Logic
 Mathematical Structures in Computer Science
, 1993
"... We present a model of classical linear logic based on the notion of strong stability that was introduced in [BE], a work about sequentiality written jointly with Antonio Bucciarelli. ..."
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Cited by 59 (8 self)
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We present a model of classical linear logic based on the notion of strong stability that was introduced in [BE], a work about sequentiality written jointly with Antonio Bucciarelli.
A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 53 (8 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
Linear Logic Without Boxes
, 1992
"... Girard's original definition of proof nets for linear logic involves boxes. The box is the unit for erasing and duplicating fragments of proof nets. It imposes synchronization, limits sharing, and impedes a completely local view of computation. Here we describe an implementation of proof nets withou ..."
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Cited by 53 (0 self)
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Girard's original definition of proof nets for linear logic involves boxes. The box is the unit for erasing and duplicating fragments of proof nets. It imposes synchronization, limits sharing, and impedes a completely local view of computation. Here we describe an implementation of proof nets without boxes. Proof nets are translated into graphs of the sort used in optimal calculus implementations; computation is performed by simple graph rewriting. This graph implementation helps in understanding optimal reductions in the calculus and in the various programming languages inspired by linear logic. 1 Beyond the calculus The calculus is not entirely explicit about the operations of erasing and duplicating arguments. These operations are important both in the theory of the  calculus and in its implementations, yet they are typically treated somewhat informally, implicitly. The proof nets of linear logic [1] provide a refinement of the calculus where these operations become explici...
Focusing and Polarization in Linear, Intuitionistic, and Classical Logics
, 2009
"... A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitioni ..."
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Cited by 43 (18 self)
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A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.
CutElimination and a PermutationFree Sequent Calculus for Intuitionistic Logic
, 1998
"... We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ..."
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Cited by 40 (6 self)
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We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.