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Axiomatics for Data Refinement in Call By Value Programming Languages
"... We give a systematic category theoretic axiomatics for modelling data refinement in call by value programming languages. Our leading examples of call by value languages are extensions of the computational calculus, such as FPC and languages for modelling nondeterminism, and extensions of the first ..."
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We give a systematic category theoretic axiomatics for modelling data refinement in call by value programming languages. Our leading examples of call by value languages are extensions of the computational calculus, such as FPC and languages for modelling nondeterminism, and extensions of the first order fragment of the computational calculus, such as a CPS language. We give a category theoretic account of the basic setting, then show how to model contexts, then arbitrary type and term constructors, then signatures, and finally data refinement. This extends and clarifies Kinoshita and Power's work on lax logical relations for call by value languages.
A Compositional Generalisation of Logical Relations
, 1998
"... Binary logical relations do not compose. So we generalise the notion of logical relation to one of lax logical relation, so that binary lax logical relations do compose. We give both a direct generalisation and a corresponding category theoretic formulation. We generalise the Basic Lemma for logical ..."
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Binary logical relations do not compose. So we generalise the notion of logical relation to one of lax logical relation, so that binary lax logical relations do compose. We give both a direct generalisation and a corresponding category theoretic formulation. We generalise the Basic Lemma for logical relations to a Basic Lemma for lax logical relations. Finally, we give an axiomatic category theoretic analysis of our definition.
Specification Refinement with System F, The HigherOrder Case
, 2000
"... . A typetheoretic counterpart to the notion of algebraic specification refinement is discussed for abstract data types with higherorder signatures. The typetheoretic setting consists of System F and the logic for parametric polymorphism of Plotkin and Abadi. For firstorder signatures, this setti ..."
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. A typetheoretic counterpart to the notion of algebraic specification refinement is discussed for abstract data types with higherorder signatures. The typetheoretic setting consists of System F and the logic for parametric polymorphism of Plotkin and Abadi. For firstorder signatures, this setting immediately gives a natural notion of specification refinement up to observational equivalence via the notion of simulation relation. Moreover, a proof strategy for proving observational refinements formalised by Bidoit, Hennicker and Wirsing can be soundly imported into the type theory. In lifting these results to the higherorder case, we find it necessary firstly to develop an alternative simulation relation and secondly to extend the parametric PERmodel interpretation, both in such a way as to observe data type abstraction barriers more closely. 1 Introduction One framework in algebraic specification that has particular appeal and applicability is that of stepwise specification refi...
A Generalisation of Prelogical Predicates to Simply Typed Formal Systems
 In ICALP
, 2004
"... We generalise the notion of prelogical predicates [HS02] to arbitrary simply typed formal systems and their categorical models. We establish the basic lemma of prelogical predicates and composability of binary prelogical relations in this generalised setting. This generalisation takes place in ..."
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We generalise the notion of prelogical predicates [HS02] to arbitrary simply typed formal systems and their categorical models. We establish the basic lemma of prelogical predicates and composability of binary prelogical relations in this generalised setting. This generalisation takes place in a categorical framework for typed higherorder abstract syntax and semantics [Fio02,MS03].
Logical Relations and Parametricity A Reynolds Programme for Category Theory and Programming Languages
, 2013
"... Dedicated to the memory of John C. Reynolds, 19352013 In his seminal paper on “Types, Abstraction and Parametric Polymorphism, ” John Reynolds called for homomorphisms to be generalized from functions to relations. He reasoned that such a generalization would allow typebased “abstraction ” (repre ..."
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Dedicated to the memory of John C. Reynolds, 19352013 In his seminal paper on “Types, Abstraction and Parametric Polymorphism, ” John Reynolds called for homomorphisms to be generalized from functions to relations. He reasoned that such a generalization would allow typebased “abstraction ” (representation independence, information hiding, naturality or parametricity) to be captured in a mathematical theory, while accounting for higherorder types. However, after 30 years of research, we do not yet know fully how to do such a generalization. In this article, we explain the problems in doing so, summarize the work carried out so far, and call for a renewed attempt at addressing
Objects and Classes in Algollike Languages 1
"... Many objectoriented languages used in practice descend from Algol. With this motivation, we study the theoretical issues underlying such languages via the theory of Algollike languages. It is shown that the basic framework of this theory extends cleanly and elegantly to the concepts of objects an ..."
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Many objectoriented languages used in practice descend from Algol. With this motivation, we study the theoretical issues underlying such languages via the theory of Algollike languages. It is shown that the basic framework of this theory extends cleanly and elegantly to the concepts of objects and classes. Moreover, a clear correspondence emerges between classes and abstract data types, whose theory corresponds to that of existential types. Equational and Hoarelike reasoning methods, and relational parametricity provide powerful formal tools for reasoning about Algollike objectoriented programs. Key Words: Algollike languages, relational parametricity, specication logic, objectoriented programming, semantics.
ENRICHED LAWVERE THEORIES Dedicated to
"... ABSTRACT. We dene the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally nitely presentable as a closed category. ..."
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ABSTRACT. We dene the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally nitely presentable as a closed category.
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"... cfl 2000 John Wiley & Sons Many objectoriented languages used in practice descend from Algol. With this motivation, we study the theoretical issues underlying such languages via the theory of Algollike languages. It is shown that the basic framework of this theory extends cleanly and elegantly ..."
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cfl 2000 John Wiley & Sons Many objectoriented languages used in practice descend from Algol. With this motivation, we study the theoretical issues underlying such languages via the theory of Algollike languages. It is shown that the basic framework of this theory extends cleanly and elegantly to the concepts of objects and classes. Moreover, a clear correspondence emerges between classes and abstract data types, whose theory corresponds to that of existential types. Equational and Hoarelike reasoning methods, and relational parametricity provide powerful formal tools for reasoning about Algollike objectoriented programs.
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.
Sketches
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 1999
"... We generalise the notion of sketch. For any locally nitely presentable category, one can speak of algebraic structure on the category, or equivalently, a finitary monad on it. For any such finitary monad, we define the notions of sketch and strict model and prove that any sketch has a generic stri ..."
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We generalise the notion of sketch. For any locally nitely presentable category, one can speak of algebraic structure on the category, or equivalently, a finitary monad on it. For any such finitary monad, we define the notions of sketch and strict model and prove that any sketch has a generic strict model on it. This is all done with enrichment in any monoidal biclosed