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**21 - 25**of**25**### Adrian Fiech

"... We present a denotational model for F < , the extension of second-order lambda calculus with subtyping defined in [Cardelli Wegner 1985]. Types are interpreted as arbitrary cpos and elements of types as natural transformations. We prove the soundness of our model with respect to the equational theor ..."

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We present a denotational model for F < , the extension of second-order lambda calculus with subtyping defined in [Cardelli Wegner 1985]. Types are interpreted as arbitrary cpos and elements of types as natural transformations. We prove the soundness of our model with respect to the equational theory of F < [Cardelli et al. 1991] and show coherence. Our model is of independent interest, because it integrates ad-hoc and parametric polymorphism in an elegant fashion, admits nontrivial records and record update operations, and formalizes an "order faithfulness" criterion for well-behaved multiple subtyping. 0 Introduction A variety of denotational models for F < (the second--order lambda calculus with subtyping and bounded quantification) [Cardelli Wegner 1985] have already been proposed [MacQueen et al. 1986, Bruce et al. 1990, Bruce Longo 1990, Amadio 1991, Cardone 1989, Abadi Plotkin 1990, Bruce Mitchell 1992, Breazu--Tannen et al. 1991], and this paper presents yet one more. Our rea...

### Impredicative Representations of Categorical Datatypes

, 1994

"... this document that certain implications are not based on a well stated formal theory but require a certain amount of hand-waving. ..."

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this document that certain implications are not based on a well stated formal theory but require a certain amount of hand-waving.

### MFPS 2006 Bunching for Regions and Locations

"... There are a number of applied lambda-calculi in which terms and types are annotated with parameters denoting either locations or locations in machine memory. Such calculi have been designed with safe memory-management operations in mind. It is difficult to construct directly denotational models for ..."

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There are a number of applied lambda-calculi in which terms and types are annotated with parameters denoting either locations or locations in machine memory. Such calculi have been designed with safe memory-management operations in mind. It is difficult to construct directly 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 memory-management operations. Bunched polymorphism provides natural type-theoretic mechanisms for capturing the disjointness conditions in such models. We illustrate this by adding regions to the basic disjointness model of αλ, the lambda-calculus associated to the logic of bunched implications. We show how both additive and multiplicative polymorphic quantifiers arise naturally in our models. A locations model is a special case. In order to relate this enterprise back to previous work on memory-management, 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.

### Bunching for Regions and Locations

- MFPS 2006
, 2006

"... There are a number of applied lambda-calculi in which terms and types are annotated with parameters denoting either locations or locations in machine memory. Such calculi have been designed with safe memory-management operations in mind. It is difficult to construct directly denotational models for ..."

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There are a number of applied lambda-calculi in which terms and types are annotated with parameters denoting either locations or locations in machine memory. Such calculi have been designed with safe memory-management operations in mind. It is difficult to construct directly 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 memory-management operations. Bunched polymorphism provides natural type-theoretic mechanisms for capturing the disjointness conditions in such models. We illustrate this by adding regions to the basic disjointness model of αλ, the lambda-calculus associated to the logic of bunched implications. We show how both additive and multiplicative polymorphic quantifiers arise naturally in our models. A locations model is a special case. In order to relate this enterprise back to previous work on memory-management, 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.

### TYPES, SETS AND CATEGORIES

"... This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category t ..."

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This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, I have elected to offer detailed technical presentations of just a few important instances. 1 THE ORIGINS OF TYPE THEORY The roots of type theory lie in set theory, to be precise, in Bertrand Russell’s efforts to resolve the paradoxes besetting set theory at the end of the 19 th century. In analyzing these paradoxes Russell had come to find the set, or class, concept itself philosophically perplexing, and the theory of types can be seen as the outcome of his struggle to resolve these perplexities. But at first he seems to have regarded type theory as little more than a faute de mieux.