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Polymorphic Linear Logic and Topos Models
 Math. Reports, Academy of Science (Canada) XII
, 1990
"... We give a definition of a "linear fibration", which is a hyperdoctrine model of polymorphic linear logic, and show how to internalise the fibration, generating topos models. This gives a constructive set theoretical context for the logic of Petri nets, as recently developed by N. Mart&apos ..."
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We give a definition of a "linear fibration", which is a hyperdoctrine model of polymorphic linear logic, and show how to internalise the fibration, generating topos models. This gives a constructive set theoretical context for the logic of Petri nets, as recently developed by N. Mart'iOliet and J. Meseguer. Also, we sketch how this can be further extended to include the exponential operator ! . In this context, the topos model we construct can be embedded in the model constructed by A.M. Pitts. 0 Introduction In [4], it is shown how to enrich the logic of Petri nets with gedanken states and processes, by embedding it into linear logic. Recently, Mart'iOliet and Meseguer have asked how to extend this even further to include polymorphism. It turns out that the process is fairly straightforward, and for maximal impact (so as to include constructive set theory), can be internalised to give topos models of polymorphic linear logic. Here we give the necessary definitions, and sketch the ...
Categorical programming for data types with restricted parametricity. Manuscript, submitted to PostProc
 of TFP
"... Abstract. Many concepts from category theory have proven useful as tools for program abstraction, particularly in functional programming. For example, many parametric data types admit operations defining a functor and related structures such as a monad. However, some parametric data types whose oper ..."
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Abstract. Many concepts from category theory have proven useful as tools for program abstraction, particularly in functional programming. For example, many parametric data types admit operations defining a functor and related structures such as a monad. However, some parametric data types whose operations are restricted in their parametricity are not amenable to traditional categorytheoretic abstractions in Haskell, despite appearing to satisfy the mathematical definitions. This paper explains the limitations of various traditional categorytheoretic approaches in Haskell, giving a precise account of their categorytheoretic meaning and the categorical implications of restricted parametricity arising from adhoc polymorphism provided by type classes in Haskell. Following from this analysis, we generalise Haskell’s notions of functors, monads and comonads, making use of GHC’s new constraint kinds extension, providing techniques for structuring programs with both unrestricted and restricted polymorphism.
MCS8219196, U.S. Army grants DAA2984K0061, DAA298490027. Extensional Models for Polymorphism
, 1988
"... This is a slightly revised version of MSCIS8775lLINC LAB 81 ..."
Summary
, 1993
"... In this thesis we carry out a detailed study of the (propositional) intuitionistic fragment of Girard’s linear logic (ILL). Firstly we give sequent calculus, natural deduction and axiomatic formulations of ILL. In particular our natural deduction is different from others and has important properties ..."
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In this thesis we carry out a detailed study of the (propositional) intuitionistic fragment of Girard’s linear logic (ILL). Firstly we give sequent calculus, natural deduction and axiomatic formulations of ILL. In particular our natural deduction is different from others and has important properties, such as closure under substitution, which others lack. We also study the process of reduction in all three logical formulations, including a detailed proof of cut elimination. Finally, we consider translations between Intuitionistic Logic (IL) and ILL. We then consider the linear term calculus, which arises from applying the CurryHoward correspondence to the natural deduction formulation. We show how the various proof theoretic formulations suggest reductions at the level of terms. The properties of strong normalization and confluence are proved for these reduction rules. We also consider mappings between the extended λcalculus and the linear term calculus. Next we consider a categorical model for ILL. We show how by considering the linear term calculus as an equational logic, we can derive a model: a Linear category. We consider two alternative models: firstly, one due to Seely and then one due to Lafont. Surprisingly, we find that Seely’s model is not sound, in that equal terms are not modelled with equal morphisms. We show how after adapting Seely’s model (by viewing it in a more abstract setting) it becomes a particular instance
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"... There are a number of applied lambdacalculi in which terms and types are annotated with parameters denoting either regions or locations in machine memory. Such calculi have been designed with safe memorymanagement operations in mind. It is difficult to directly construct denotational models for ex ..."
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There are a number of applied lambdacalculi in which terms and types are annotated with parameters denoting either regions or locations in machine memory. Such calculi have been designed with safe memorymanagement operations in mind. It is difficult to directly construct denotational models for existing calculi of this kind. We approach the problem differently, by starting from a class of mathematical models that describe some of the essential semantic properties intended in these calculi. In particular, disjointness conditions between regions (or locations) are implicit in many of the memorymanagement operations. Bunched polymorphism provides natural typetheoretic mechanisms for capturing the disjointness conditions in such models. We illustrate this by extending the Basic Disjointness Model of αλ with regions. We show how additive and multiplicative polymorphic quantifiers are interpreted. A locations model is a special case. In order to relate this enterprise back to previous work on memorymanagement, we provide an example in which the model is refined and used to provide a denotational semantics for a language with explicit allocation and disposal of regions. 1
Towards a Categorical Understanding of PlotkinAbadi Logic
, 1998
"... The idea of parametric polymorphism is that of a single operator that can be used for di erent data types and whose behaviour is somehow uniform for each type. This concept was rst proposed by ..."
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The idea of parametric polymorphism is that of a single operator that can be used for di erent data types and whose behaviour is somehow uniform for each type. This concept was rst proposed by
A Model for Formal Parametric Polymorphism: a Per Interpretation for System R
, 1995
"... System R is an extension of system F that formalizes Reynolds' notion of relational parametricity. In this paper we describe a semantics for system R. As a first step, we give a careful and general reconstruction of the per model of system F of Bainbridge et al., presenting it as a categorical ..."
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System R is an extension of system F that formalizes Reynolds' notion of relational parametricity. In this paper we describe a semantics for system R. As a first step, we give a careful and general reconstruction of the per model of system F of Bainbridge et al., presenting it as a categorical model in the sense of Seely. Then we interpret system R in this model. LIENS, CNRS  D'epartement de Math'ematiques et Informatique de l'Ecole Normale Sup'erieure and Dipartimento di Matematica, Universit`a di Siena, Italy y Digital Equipment Corporation, Systems Research Center z LIENS, CNRS  D'epartement de Math'ematiques et Informatique de l'Ecole Normale Sup'erieure Contents 1 From models of F to models of R 3 2 System R 4 3 Categories of pers and categories of relations 6 3.1 Combinatory algebras and models : : : : : : : : : : : : : : : : : : : : 7 3.2 Categories PER and SAT : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3.3 Constructions in SAT : : : : : : : : : : : : :...