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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 ..."
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Cited by 101 (4 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.
An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory
 In Proceedings of CSL 96
, 1997
"... . We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of ..."
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Cited by 17 (6 self)
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. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !inductive fixedpoint object (Definition 1...
A Mixed Linear and NonLinear Logic: Proofs, Terms and Models (Extended Abstract)\Lambda
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Linear Types and Approximation
, 1994
"... We enrich the autonomous category of complete lattices and maps preserving all suprema with the important concept of approximation by specifying a autonomous full subcategory LFS of linear FSlattices . This is the greatest autonomous full subcategory of linked bicontinuous lattices. The modalit ..."
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We enrich the autonomous category of complete lattices and maps preserving all suprema with the important concept of approximation by specifying a autonomous full subcategory LFS of linear FSlattices . This is the greatest autonomous full subcategory of linked bicontinuous lattices. The modalities !() and ?() mediate a duality between the upper and lower powerdomains. The distributive objects in LFS give rise to the compact closed autonomous full subcategory CD of completely distributive lattices . We characterize algebraic objects in LFS by forbidden substructures "a la Plotkin'. Keywords: autonomous category, linear logic, interaction orders, bicontinuous lattices, completely distributive lattices, upper and lower powerdomains. 1 Introduction Complete lattices with maps preserving all suprema as morphisms form a autonomous category SUP [Bar79] and give rise to a model of linear logic in the standard fashion of [See89]. If 2 := f0 ! 1g denotes the twopoint lattice and A...