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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 ..."
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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, 19 ..."
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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.
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"... We study a weakening of the notion of logical relations, called prelogical relations, that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of ..."
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We study a weakening of the notion of logical relations, called prelogical relations, that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of logical relations to hold only for pairs of functions that are expressible by the same lambda term. Prelogical relations are the minimal weakening of logical relations that gives composability for extensional structures and simultaneously the most liberal definition that gives the Basic Lemma. Prelogical predicates (i.e., unary prelogical relations) coincide with sets that are invariant under Kripke logical relations with varying arity as introduced by Jung and Tiuryn, and prelogical relations are the closure under projection and intersection of logical relations. These conceptually independent characterizations of prelogical relations suggest that the concept is rather intrinsic and robust. The use of prelogical relations gives an improved version of Mitchellâ€™s representation independence theorem which characterizes observational equivalence for all signatures rather than just for firstorder signatures. Prelogical relations can be used in place of logical relations to give an account of data refinement where the fact that prelogical relations compose explains why stepwise refinement is sound.