Results 1  10
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10
Positive Subtyping
 Information and Computation
, 1994
"... The statement S T in a calculus with subtyping is traditionally interpreted as a semantic coercion function of type [[S]]![[T ]] that extracts the "T part" of an element of S. If the subtyping relation is restricted to covariant positions, this interpretation may be enriched to include both the ..."
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Cited by 51 (8 self)
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The statement S T in a calculus with subtyping is traditionally interpreted as a semantic coercion function of type [[S]]![[T ]] that extracts the "T part" of an element of S. If the subtyping relation is restricted to covariant positions, this interpretation may be enriched to include both the coercion and an overwriting function put[S; T ] 2 [[S]]![[T ]]![[S]] that updates the T part of an element of S.
Decidability of HigherOrder Subtyping with Intersection Types
 University of Edinburgh, LFCS
, 1994
"... The combination of higherorder subtyping with intersection types yields a typed model of objectoriented programming with multiple inheritance [11]. The target calculus, F ! , a natural generalization of Girard's system F ! with intersection types and bounded polymorphism, is of independent i ..."
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Cited by 40 (11 self)
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The combination of higherorder subtyping with intersection types yields a typed model of objectoriented programming with multiple inheritance [11]. The target calculus, F ! , a natural generalization of Girard's system F ! with intersection types and bounded polymorphism, is of independent interest, and is our subject of study. Our main contribution is the proof that subtyping in F ! is decidable. This yields as a corollary the decidability of subtyping in F ! , its intersection free fragment, because the F ! subtyping system is a conservative extension of that of F ! . The calculus presented in [8] has no reductions on types. In the F ! subtyping system the presence of ficonversion  an extension of ficonversion with distributivity laws  drastically increases the complexity of proving the decidability of the subtyping relation. Our proof consists of, firstly, defining an algorithmic presentation of the subtyping system of F ! , secondly, proving that th...
A Unifying TypeTheoretic Framework for Objects
, 1993
"... We give a direct typetheoretic characterization of the basic mechanisms of objectoriented programming, including objects, methods, message passing, and subtyping, by introducing an explicit constructor for object types and suitable introduction, elimination, and equality rules. The resulting abst ..."
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Cited by 37 (9 self)
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We give a direct typetheoretic characterization of the basic mechanisms of objectoriented programming, including objects, methods, message passing, and subtyping, by introducing an explicit constructor for object types and suitable introduction, elimination, and equality rules. The resulting abstract framework provides a basis for justifying and comparing previous encodings of objects based on recursive record types (Cardelli, 1984; Cardelli, 1992; Bruce, 1994; Cook et al., 1990; Mitchell, 1990a) and encodings based on existential types (Pierce & Turner, 1994).
Subject reduction and minimal types for higher order subtyping
 In Proceedings of the Second Chinese Language Processing Workshop
, 1997
"... We define the typed lambda calculus F ω ∧ , a natural generalization of Girard’s system F ω with intersection types and bounded polymorphism. A novel aspect of our presentation is the use of term rewriting techniques to present intersection types, which clearly splits the computational semantics (re ..."
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Cited by 6 (3 self)
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We define the typed lambda calculus F ω ∧ , a natural generalization of Girard’s system F ω with intersection types and bounded polymorphism. A novel aspect of our presentation is the use of term rewriting techniques to present intersection types, which clearly splits the computational semantics (reduction rules) from the syntax (inference rules) of the system. We establish properties such as ChurchRosser for the reduction relation on types and terms, and Strong Normalization for the reduction on types. We prove that types are preserved by computation (Subject Reduction property), and that the system satisfies the Minimal Types property. On the way to establishing these results, we define algorithms for type inference and subtype checking. 1
Coercive Subtyping for the Calculus of Constructions (extended abstract)
"... We present a coercive subtyping system for the calculus of constructions. The proposed system λC co ≤ is obtained essentially by adding coercions and ηconversion to λC≤[10], which is a subtyping extension to the calculus of constructions without coercions. Following [17, 18], the coercive subtyping ..."
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Cited by 2 (1 self)
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We present a coercive subtyping system for the calculus of constructions. The proposed system λC co ≤ is obtained essentially by adding coercions and ηconversion to λC≤[10], which is a subtyping extension to the calculus of constructions without coercions. Following [17, 18], the coercive subtyping c: A ≤ B is understood as a special case of typing in arrow type c: A → B such that the term c behaves like an identity function. We prove that, with respect to this semantic interpretation, the proposed coercive subtyping system is sound and complete, and that this completeness leads to transitivity elimination (transitivity rule is admissible). In and CCβη, this fact implies that λC co ≤ has confluence, subject reduction and strong normalization. We propose a formalization of coercion inference problem and present a sound and complete coercion inference algorithm. addition, we establish the equivalence between λC co
Subtyping Parametric and Dependent Types
, 1996
"... A type may be a subtype of another type. The intuition about this should be clear: a type is a type of data, some data then may live in a given type as well as in a larger one, up to a simple "transformation". The advantage is that those data may be "seen" or used in different contexts. The formal ..."
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Cited by 1 (0 self)
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A type may be a subtype of another type. The intuition about this should be clear: a type is a type of data, some data then may live in a given type as well as in a larger one, up to a simple "transformation". The advantage is that those data may be "seen" or used in different contexts. The formal treatment of this intuition, though, is not so obvious, in particular when data may be programs. In Object Oriented Programming, where the issue of "reusing data" is crucial, there has been a longlasting discussion on "inheritance" and ... little agreement. There are several ways to understand and formalize inheritance, which depend on the specific programming environment used. Since early work of Cardelli and Wegner, there has been a large amount of papers developing several possible functional approaches to inheritance, as subtyping. Indeed, functional subtyping captures only one point of view on inheritance, yet this notion largely motivated most of that work. Whethe
Subtyping Parametric and Dependent Types  An introduction
, 1996
"... A type may be a subtype of another type. The intuition about this should be clear: a type is a type of data, some data then may live in a given type as well as in a larger one, up to a simple "transformation". The advantage is that those data may be "seen" or used in different contexts. The formal ..."
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A type may be a subtype of another type. The intuition about this should be clear: a type is a type of data, some data then may live in a given type as well as in a larger one, up to a simple "transformation". The advantage is that those data may be "seen" or used in different contexts. The formal treatment of this intuition, though, is not so obvious, in particular when data may be programs. In Object Oriented Programming, where the issue of "reusing data" is crucial, there has been a longlasting discussion on "inheritance" and ... little agreement. There are several ways to understand and formalize inheritance, which depend on the specific programming environment used. Since early work of Cardelli and Wegner, there has been a large amount of papers developing several possible functional approaches to inheritance, as subtyping. Indeed, functional subtyping captures only one point of view on inheritance, yet this notion largely motivated most of that work. Whethe
Dependent Type System with Subtyping  Type Level Transitivity Elimination
, 1996
"... Dependent type systems have been the basis of many proof development enviroments. In [AC96], a system P is proposed as a subtyping extension of the dependent type system P[Bar92] (also called \Pi[Dow95]). P has nice metatheoretic properties including subject reduction and decidability, but transit ..."
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Dependent type systems have been the basis of many proof development enviroments. In [AC96], a system P is proposed as a subtyping extension of the dependent type system P[Bar92] (also called \Pi[Dow95]). P has nice metatheoretic properties including subject reduction and decidability, but transitivity elimination is restricted to the fi 2 normalized types. In this report, we propose a system \Pi , which is equivalent to P , but it has type level transitivity eliminatioin property. This feature distinguishes our approach from the existing subtyping systems with reduction relations in types. e.g. P[AC96], F ! [SP94], F ! [Com94], where transitivity elimination only holds for normalized types. Metatheoretic properties including subject reduction and decidability are established. The system is shown to be equivalent with P in typing, kinding and context formation. The type checking algorithm is more clear and efficient than P . The technique is suitable for future extensions and...