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33
Notes on Sconing and Relators
, 1993
"... This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a categorytheoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. 1 Introduction Many modern programming languages feature ..."
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This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a categorytheoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. 1 Introduction Many modern programming languages feature rather sophisticated typing mechanisms. In particular, languages such as ML include polymorphic data types, which allow considerable programming flexibility. Several notions of polymorphism were introduced into computer science by Strachey [Str67], among them the important notion of parametric polymorphism. Strachey's intuitive definition is that a polymorphic function is parametric if it has a uniformly given algorithm in all types, that is, if the function's behavior is independent of the type at which the function is instantiated. Reynolds [Rey83] proposed a mathematical definition of parametric polymorphic functions by means of invariance with respect to certain relations induced by typ...
HigherOrder Intersection Types and Multiple Inheritance
, 1995
"... this paper was completed, the metatheory of this system has been studied in much greater detail by Compagnoni [ Compagnoni, 1994, Compagnoni, 1995 ] . A type system combining intersection types with a powerful form of polymorphism is of independent interest. Reynolds [ 1988 ] has argued that interse ..."
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this paper was completed, the metatheory of this system has been studied in much greater detail by Compagnoni [ Compagnoni, 1994, Compagnoni, 1995 ] . A type system combining intersection types with a powerful form of polymorphism is of independent interest. Reynolds [ 1988 ] has argued that intersection types can form the basis of elegant language designs. But his Forsythe language has only a firstorder type system, and thus lacks some of the expressive possibilities of polymorphic languages like ML. Our work represents a step toward a synthesis of these styles of language design. The following section shows some examples of multiple inheritance using a simple highlevel syntax. Section 3, the core of the paper, defines the calculus F
On functors expressible in the polymorphic typed lambda calculus
 Logical Foundations of Functional Programming
, 1990
"... This is a preprint of a paper that has been submitted to Information and Computation. ..."
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This is a preprint of a paper that has been submitted to Information and Computation.
Multiple Inheritance via Intersection Types
 UNIVERSITY OF EDINBURGH
, 1993
"... Combining intersection types with higherorder subtyping yields a typed model of objectoriented programming with multiple inheritance. Objects, message passing, subtyping, and inheritance appear as programming idioms in a typed calculus, a modelling technique that facilitates experimentation and h ..."
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Cited by 12 (4 self)
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Combining intersection types with higherorder subtyping yields a typed model of objectoriented programming with multiple inheritance. Objects, message passing, subtyping, and inheritance appear as programming idioms in a typed calculus, a modelling technique that facilitates experimentation and helps in distinguishing between essential aspects of the objectoriented style encapsulation and subtype polymorphism, which are directly reflected in the lowlevel type system  and useful but inessential programming idioms such as inheritance. The target calculus, a natural generalization of system F ! with intersection types, is of independent interest. We establish basic structural properties and give a proof of type soundness using a simple semantics based on partial equivalence relations.
A Semantics for Static Type Inference
 Information and Computation
, 1993
"... Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model classes. We show compl ..."
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Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model classes. We show completeness for Curry's system itself, relative to an extended notion of model that validates reduction but not conversion.
Typing untyped λterms, or Reducibility strikes again!
, 1995
"... It was observed by Curry that when (untyped) λterms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes ..."
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It was observed by Curry that when (untyped) λterms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having headnormal forms, can be characterized in some systems D and D. The proofs use variants of the method of reducibility. In this paper, we presenta uniform approach for proving several metatheorems relating properties ofterms and their typability in the systems D and D. Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easily rederived. We alsocharacterize the terms that have weak headnormal forms, which appears to be new. We conclude by stating a number of challenging open problems regarding possible generalizations of the realizability method.
A modular typechecking algorithm for type theory with singleton types and proof irrelevance
 IN TLCA’09, VOLUME 5608 OF LNCS
, 2009
"... ..."
A realizability interpretation of MartinLöf's type theory
"... In this paper we present a simple argument for normalization of the fragment of MartinLöf's type theory that contains the natural numbers, dependent function types and the first universe. We do this by building a realizability model of this theory which directly reflects that terms and types are ge ..."
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In this paper we present a simple argument for normalization of the fragment of MartinLöf's type theory that contains the natural numbers, dependent function types and the first universe. We do this by building a realizability model of this theory which directly reflects that terms and types are generated simultaneously.
A Model for Formal Parametric Polymorphism: A PER Interpretation for System R
, 1995
"... System R is an extension of system F that formalizes Reynolds' notion of relational parametricity. In system R, considerably more lambdaterms can be proved equal than in system F: for example, the encoded weak products of F are strong products in R. Also, many "theorems for free" à la Wadler can be ..."
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Cited by 6 (0 self)
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System R is an extension of system F that formalizes Reynolds' notion of relational parametricity. In system R, considerably more lambdaterms can be proved equal than in system F: for example, the encoded weak products of F are strong products in R. Also, many "theorems for free" à la Wadler can be proved formally in R. In this paper we describe a semantics for system R. As a first step, we give a precise and general reconstruction of the per model of system F of Bainbridge et al., presenting it as a categorical model in the sense of Seely. Then we interpret system R in this model.