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A Mixed Linear and NonLinear Logic: Proofs, Terms and Models (Preliminary Report)
, 1994
"... Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (`of course') modality. Benton, Bierman, Hyland and de Paiva have given a term assignment system for ILL and an associated notion of categorical model in which the ! modality is modelled by a comonad satis ..."
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Cited by 96 (3 self)
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Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (`of course') modality. Benton, Bierman, Hyland and de Paiva have given a term assignment system for ILL and an associated notion of categorical model in which the ! modality is modelled by a comonad satisfying certain extra conditions. Ordinary intuitionistic logic is then modelled in a cartesian closed category which arises as a full subcategory of the category of coalgebras for the comonad. This paper attempts to explain the connection between ILL and IL more directly and symmetrically by giving a logic, term calculus and categorical model for a system in which the linear and nonlinear worlds exist on an equal footing, with operations allowing one to pass in both directions. We start from the categorical model of ILL given by Benton, Bierman, Hyland and de Paiva and show that this is equivalent to having a symmetric monoidal adjunction between a symmetric monoidal closed category and a cartesian closed category. We then derive both a sequent calculus and a natural deduction presentation of the logic corresponding to the new notion of model.
Lightweight linear types in system F o
 In TLDI
, 2010
"... We present System F ◦ , an extension of System F that uses kinds to distinguish between linear and unrestricted types, simplifying the use of linearity for generalpurpose programming. We demonstrate through examples how System F ◦ can elegantly express many useful protocols, and we prove that any p ..."
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Cited by 5 (1 self)
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We present System F ◦ , an extension of System F that uses kinds to distinguish between linear and unrestricted types, simplifying the use of linearity for generalpurpose programming. We demonstrate through examples how System F ◦ can elegantly express many useful protocols, and we prove that any protocol representable as a DFA can be encoded as an F ◦ type. We supply mechanized proofs of System F ◦ ’s soundness and parametricity properties, along with a nonstandard operational semantics that formalizes common intuitions about linearity and aids in reasoning about protocols. We compare System F ◦ to other linear systems, noting that the simplicity of our kindbased approach leads to a more explicit account of what linearity is meant to capture, allowing otherwiseconflicting interpretations of linearity (in particular, restrictions on aliasing versus restrictions on resource usage) to coexist peacefully. We also discuss extensions to System F ◦ aimed at making the core language more practical, including the additive fragment of linear logic, algebraic datatypes, and recursion.
Categorical Proof Theory of CoIntuitionistic Linear Logic
"... Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of ..."
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Cited by 1 (1 self)
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Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponent!, we build models of cointuitionistic logic in symmetric monoidal closed categories with additional structure, using a variant of Crolard’s term assignment to cointuitionistic logic in the construction of a free category. 1
Down With the Bureaucracy of Syntax!
"... to proof trees for classical linear logic, which bears a close resemblance to the way that pattern matching is used in programming languages. It equates the same proofs that are equated by proof nets, in the formulation of proof nets introduced by Dominic Hughes and Rob van Glabbeek; and goes beyond ..."
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to proof trees for classical linear logic, which bears a close resemblance to the way that pattern matching is used in programming languages. It equates the same proofs that are equated by proof nets, in the formulation of proof nets introduced by Dominic Hughes and Rob van Glabbeek; and goes beyond that formulation in handling exponentials and units. It provides a symmetric treatment of all the connectives, and may provide programmers with improved insight into connectives such as "par" and "why not" that are difficult to treat in programming languages based on an intuitionistic formulation of linear logic.