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63
Parametric and TypeDependent Polymorphism
, 1995
"... Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types and between types. R ..."
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Cited by 10 (5 self)
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Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types and between types. Reynolds's approach set the basis for most of the current work on parametricity, as we will review below (.3). Some twelve years earlier, Girard had given just a simple hint towards another understanding of the properties of "computing with types". In [Gir71], it is shown, as a side remark, that, given a type A, if one defines a term J A such that, for any type B, J A B reduces to 1, if A = B, and reduces to 0, if A ¹ B, then F + J A does not normalize. In particular, then, J A is not definable in F. This remark on how terms may depend on types is inspired by a view of types which is quite different from Reynolds's. System F was born as the theory of proofs of second order intuitionis...
Coherence and Transitivity of Subtyping as Entailment
, 1996
"... The relation of inclusion between types has been suggested by the practice of programming as it enriches the polymorphism of functional languages. We propose a simple (and linear) sequent calculus for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subty ..."
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The relation of inclusion between types has been suggested by the practice of programming as it enriches the polymorphism of functional languages. We propose a simple (and linear) sequent calculus for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful sequents. In particular, transitivity and antisymmetry will be derived from elementary logical principles.
The Sreplete construction
 In CTCS 55, pages 96  116. Springer Lecture Notes in Computer Science 953
, 1995
"... this paper: (internal version) if C 1 is a quasitopos, then S ..."
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Cited by 9 (2 self)
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this paper: (internal version) if C 1 is a quasitopos, then S
The Modified Realizability Topos
 Journal of Pure and Applied Algebra
, 1996
"... The modified realizability topos is the semantic (and higher order) counterpart of a variant of Kreisel's modified realizability (1957). These years, this realizability has been in the limelight again because of its possibilities for modelling type theory (Streicher, HylandOngRitter) and stron ..."
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Cited by 9 (1 self)
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The modified realizability topos is the semantic (and higher order) counterpart of a variant of Kreisel's modified realizability (1957). These years, this realizability has been in the limelight again because of its possibilities for modelling type theory (Streicher, HylandOngRitter) and strong normalization.
Distinguishing Data Structures and Functions: the Constructor Calculus and Functorial Types
 Typed Lambda Calculi and Applications: 5th International Conference TLCA 2001
, 2001
"... The expressive power of functional programming can be improved by identifying and exploiting the characteristics that distinguish data types from function types. Data types support generic functions for equality, mapping, folding, etc. that do not apply to functions. Such generic functions requi ..."
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Cited by 9 (1 self)
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The expressive power of functional programming can be improved by identifying and exploiting the characteristics that distinguish data types from function types. Data types support generic functions for equality, mapping, folding, etc. that do not apply to functions. Such generic functions require case analysis, or patternmatching, where the branches may have incompatible types, e.g. products or sums. This is handled in the constructor calculus where specialisation of program extensions is governed by constructors for data types. Typing of generic functions employs polymorphism over functors in a functorial type system. The expressive power is greatly increased by allowing the functors to be polymorphic in the number of arguments they take, i.e. in their arities. The resulting system can define and type the fundamental examples above. Some basic properties are established, namely subject reduction, the ChurchRosser property, and the existence of a practical type inference algorithm.
Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory
 In Proc. 17th IEEE Symposium on Logic in Computer Science
, 2003
"... This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown ..."
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Cited by 9 (2 self)
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This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic ZermeloFraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambdacalculus with callbyvalue operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domaintheoretic models.
Axioms and (Counter)examples in Synthetic Domain Theory
 Annals of Pure and Applied Logic
, 1998
"... this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the p ..."
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Cited by 8 (7 self)
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this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the purposes of the axiomatic part of this paper, we believe that it would also be
Data Categories
 Computing: The Australasian Theory Symposium Proceedings
, 1996
"... Data categories and functors, and the strong natural transformations between them provide a universe in which to model parametric polymorphism. Data functors are distinguished by being decomposable into shape and data, i.e. they represent types that store data. Every strong transformation between tw ..."
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Cited by 7 (5 self)
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Data categories and functors, and the strong natural transformations between them provide a universe in which to model parametric polymorphism. Data functors are distinguished by being decomposable into shape and data, i.e. they represent types that store data. Every strong transformation between two such is given by a uniform algorithm, and so may represent a polymorphic term. The data functors are closed under composition, finite products and sums, exponentiation by an object, final coalgebras and initial algebras. For any two such, the collection of strong natural transformations between them is representable by an object. The covariant type system supports parametric polymorphism on data types, and can be modelled in a data category. Since the category of sets is a data category, it follows that parametric polymorphism can have a settheoretic model. Keywords data categories covariance parametric polymorphism. 1 Introduction This paper introduces data functors, the data categor...
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...