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Operational Properties of Lily, a Polymorphic Linear Lambda Calculus with Recursion
"... Plotkin has advocated the combination of linear lambda calculus, polymorphism and fixed point recursion as an expressive semantic metalanguage. We study its expressive power from an operational point of view. We show that the naturally callbyvalue operators of linear lambda calculus can be given a ..."
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Cited by 35 (1 self)
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Plotkin has advocated the combination of linear lambda calculus, polymorphism and fixed point recursion as an expressive semantic metalanguage. We study its expressive power from an operational point of view. We show that the naturally callbyvalue operators of linear lambda calculus can be given a callbyname semantics without affecting termination at exponential types and hence without affecting ground contextual equivalence. This result is used to prove properties of a logical relation that provides a new extensional characterisation of ground contextual equivalence and relational parametricity properties of polymorphic types.
From Action Calculi to Linear Logic
, 1998
"... . Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give ..."
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Cited by 19 (7 self)
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. Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Benton's models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a modelembedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi. 1 Introduction Action calculi arose directly from the ßcalculus [MPW92]. They were introduced by Milner [Mil96], to provide a uniform notation for capturing many calculi of interaction such as the ßcalculus, the calculus, models of distribut...
On an Intuitionistic Modal Logic
 Studia Logica
, 2001
"... . In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models ..."
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Cited by 19 (4 self)
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. In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models of IS4, not in the framework of Kripke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability. 1. Introduction Modal logics are traditionally extensions of classical logic with new operators, or modalities, whose operation is intensional. Modal logics are most commonly justified by the provision of an intuitive semantics based upon `possible worlds', an idea originally due to Kripke. Kripke also provided a possible worlds semantics for intuitionistic logic, and so it is natural to consider intuitionistic logic extended with intensional modalities...
Relating Categorical Semantics for Intuitionistic Linear Logic
 APPLIED CATEGORICAL STRUCTURES
, 2005
"... There are several kinds of linear type theory in the literature, some with their associated notion of categorical model. Our aim on this paper is to systematize the relationships amongst three of these linear type theories and their models. We point out that mere soundness and completeness of a line ..."
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Cited by 14 (2 self)
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There are several kinds of linear type theory in the literature, some with their associated notion of categorical model. Our aim on this paper is to systematize the relationships amongst three of these linear type theories and their models. We point out that mere soundness and completeness of a linear type theory with respect to a class of categorical models is not sufficient to uniquely identify the most appropriate class of models for the theory in question. We recommend instead the use of internal languages. Considering their internal languages we relate the categories of models in the literature via reflections and coreections.
Linear Explicit Substitutions
 In Proc. of Westapp'98
, 1998
"... The oecalculus adds explicit substitutions to the calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the oecalculus to provide a linear calculus of explicit substitutions, called xDILL, which ..."
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Cited by 12 (8 self)
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The oecalculus adds explicit substitutions to the calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the oecalculus to provide a linear calculus of explicit substitutions, called xDILL, which analogously describes the implementation of linear functional programming languages. Our main observation is that there are nontrivial interactions between linearity and explicit substitutions and that xDILL is therefore best understood as a synthesis of its underlying logical structure and the technology of explicit substitutions. This is in contrast to the oecalculus where the explicit substitutions are independent of the underlying logical structure. Keywords: calculus, explicit substitutions, linear logic 1 Introduction This paper combines the technologies of explicit substitutions and linearity in a mathematically consistent way. We start by describing these technologies and the...
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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Cited by 10 (4 self)
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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
Categorical and domain theoretic models of parametric polymorphism
, 2005
"... We present a domaintheoretic model of parametric polymorphism based on admissible per’s over a domaintheoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPLstructure as defined by the authors in ..."
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Cited by 9 (6 self)
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We present a domaintheoretic model of parametric polymorphism based on admissible per’s over a domaintheoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPLstructure as defined by the authors in [7, 5]. This construction gives formal proof of solutions to a large class of recursive domain equations, which we explicate. As an example of a computation in the model, we explicitly describe the natural numbers object obtained using parametricity. The theory of admissible per’s can be considered a domain theory for (impredicative) polymorphism. By studying various categories of admissible and chain complete per’s and their relations, we discover a picture very similar to that of domain theory. 1
Kripke Resource Models of a DependentlyTyped, Bunched lambdaCalculus (Extended Abstract)
, 1999
"... The lLcalculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Second ..."
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Cited by 8 (6 self)
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The lLcalculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proofobjects of BI, the logic of bunched implications. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intuitionistic quantifiers. The lLcalculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we describe the categorical semantics of the lLcalculus. This is given by Kripke resource models, which are monoidindexed sets of functorial Kripke models, the monoid giving an account of resource consumption. We describe a class of concrete, settheoretic models. The models are given by the category of families of sets, parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.
Enriching an Effect Calculus with Linear Types
"... Abstract. We define an enriched effect calculus by extending a type theory for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and ..."
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Cited by 8 (4 self)
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Abstract. We define an enriched effect calculus by extending a type theory for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and/or continuations. Our main syntactic result is the conservativity of the enriched effect calculus over a basic effect calculus without linear primitives (closely related to Moggi’s computational metalanguage, Filinski’s effect PCF and Levy’s callbypushvalue). The proof of this syntactic theorem makes essential use of a categorytheoretic semantics, whose study forms the second half of the paper. Our semantic results include soundness, completeness, the initiality of a syntactic model, and an embedding theorem: every model of the basic effect calculus fully embeds in a model of the enriched calculus. The latter means that our enriched effect calculus is applicable to arbitrary computational effects, answering in the positive a question of Benton and Wadler (LICS 1996). 1