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Prelogical Relations
, 1999
"... this paper but which have some intriguing connections to some of our results and techniques, are [32] and [20]. We believe that the concept of prelogical relation would have a beneficial impact on the presentation and understanding of their results ..."
Abstract

Cited by 26 (5 self)
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this paper but which have some intriguing connections to some of our results and techniques, are [32] and [20]. We believe that the concept of prelogical relation would have a beneficial impact on the presentation and understanding of their results
On PlotkinAbadi Logic for Parametric Polymorphism  Towards a Categorical Understanding
"... . The idea of parametric polymorphism is that of a single operator that can be used for different data types and whose behaviour is somehow uniform for each type. Reynolds [Reynolds, 1983] uses binary relations to define a uniformity condition for parametric polymorphism. In [Plotkin & Abadi, 1993] ..."
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. The idea of parametric polymorphism is that of a single operator that can be used for different data types and whose behaviour is somehow uniform for each type. Reynolds [Reynolds, 1983] uses binary relations to define a uniformity condition for parametric polymorphism. In [Plotkin & Abadi, 1993] the authors proposed a second order logic for second order lambdacalculus; this logic is able to handle parametric polymorphism in the binary relational sense of Reynolds. In this paper we examine a categorical framework for this logic. This framework is based on the notion of categorical model of second order lambdacalculus as given, for example, in [Pitts, 1987, Seely, 1987, Robinson, 1992, Jacobs, 1991]. Going through the categorical constructions of the model, an unexpected property of quantification over type variables appears. A simple categorical calculation indicates what is the appropriate way to obtain the right adjoint to weakening that models universal quantification. The resul...
Logical Relations, Data Abstraction, and Structured Fibrations
"... We develop a notion of equivalence between interpretations of the simply typed calculus together with an equationally dened abstract datatype, and we show that two interpretations are equivalent if and only if they are linked by a logical relation. We show that our construction generalises from th ..."
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We develop a notion of equivalence between interpretations of the simply typed calculus together with an equationally dened abstract datatype, and we show that two interpretations are equivalent if and only if they are linked by a logical relation. We show that our construction generalises from the simply typed calculus to include the linear calculus and calculi with additional type and term constructors, such as those given by sum types or by a strong monad for modelling phenomena such as partiality or nondeterminism. This is all done in terms of category theoretic structure, using  brations to model logical relations following Hermida, and adapting Jung and Tiuryn's logical relations of varying arity to provide the completeness results, which form the heart of the work.