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Programming with Intersection Types and Bounded Polymorphism
, 1991
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representing the official policies, either expressed or implied, of the U.S. Government.
Intersection Types and Bounded Polymorphism
, 1996
"... this paper (Compagnoni, Intersection Types and Bounded Polymorphism 3 1994; Compagnoni, 1995) has been used in a typetheoretic model of objectoriented multiple inheritance (Compagnoni & Pierce, 1996). Related calculi combining restricted forms of intersection types with higherorder polymorph ..."
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this paper (Compagnoni, Intersection Types and Bounded Polymorphism 3 1994; Compagnoni, 1995) has been used in a typetheoretic model of objectoriented multiple inheritance (Compagnoni & Pierce, 1996). Related calculi combining restricted forms of intersection types with higherorder polymorphism and dependent types have been studied by Pfenning (Pfenning, 1993). Following a more detailed discussion of the pure systems of intersections and bounded quantification (Section 2), we describe, in Section 3, a typed calculus called F ("Fmeet ") integrating the features of both. Section 4 gives some examples illustrating this system's expressive power. Section 5 presents the main results of the paper: a prooftheoretic analysis of F 's subtyping and typechecking relations leading to algorithms for checking subtyping and for synthesizing minimal types for terms. Section 6 discusses semantic aspects of the calculus, obtaining a simple soundness proof for the typing rules by interpreting types as partial equivalence relations; however, another prooftheoretic result, the nonexistence of least upper bounds for arbitrary pairs of types, implies that typed models may be more difficult to construct. Section 7 offers concluding remarks. 2. Background
Rank 2 Type Systems and Recursive Definitions
, 1995
"... We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We int ..."
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We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We introduce a rank 2 system combining intersections and polymorphism, and prove that it types exactly the same terms as the other rank 2 systems. The combined system suggests a new rule for typing recursive definitions. The result is a rank 2 type system with decidable type inference that can type some interesting examples of polymorphic recursion. Finally,we discuss some applications of the type system in data representation optimizations such as unboxing and overloading.
A New Presentation of the Intersection Type Discipline Through Principal Typings of Normal Forms
, 1996
"... We introduce an intersection type system which is a restriction of the intersection type discipline. This restriction leads to a principal type property for normal forms in the classical sense, while retaining the expressivity of the classical discipline. We characterize the structure of principal ..."
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We introduce an intersection type system which is a restriction of the intersection type discipline. This restriction leads to a principal type property for normal forms in the classical sense, while retaining the expressivity of the classical discipline. We characterize the structure of principal types of normal forms and give an algorithm that reconstructs normal forms from types. Having shown the equivalence between principal types and normal forms, we define an expansion operation on types which allows us to recover all possible types for any normalizable term. The contribution of this work is a new and simpler presentation of the intersection type discipline through a purely syntactic and completely characterized notion of principal types.
Haskellstyle Overloading is NPhard
 In Proceedings of the 1994 International Conference on Computer Languages
, 1994
"... Extensions of the ML type system, based on constrained type schemes, have been proposed for languages with overloading. Type inference in these systems requires solving the following satisfiability problem. Given a set of type assumptions C over finite types and a type basis A, is there is a substit ..."
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Extensions of the ML type system, based on constrained type schemes, have been proposed for languages with overloading. Type inference in these systems requires solving the following satisfiability problem. Given a set of type assumptions C over finite types and a type basis A, is there is a substitution S that satisfies C in that A ` CS is derivable? Under arbitrary overloading, the problem is undecidable. Haskell limits overloading to a form similar to that proposed by Kaes called parametric overloading. We formally characterize parametric overloading in terms of a regular tree language and prove that although decidable, satisfiability is NPhard when overloading is parametric. 1 Introduction A practical limitation of the ML type system is that it prohibits global overloading in a programming language by restricting to at most one the number of assumptions per identifier in a type context, a limitation noted by Milner himself [Mil78]. Suppose we wish to assert that a free identifier...
A Note on Intersection Types
, 1995
"... : Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating th ..."
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: Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating that any term which possesses a type in D strongly normalises does not need a new reducibility argument, but is a mere consequence of strong normalization for natural deduction restricted to the conjunction and implication. The proof of strong normalization for natural deduction, and therefore our result, as opposed to reducibility arguments, can be carried out within primitive recursive arithmetic. On the other hand, this enlightens the relation between & and & that G. Pottinger has already wondered about, and can be applied to other situations, like the lambda calculus with multiplicities of G. Boudol. Keywords: Lambda calculus , intersection types , strong normalization. Logic, proof th...
Direct Proofs of Strong Normalisation in Calculi of Explicit Substitutions
"... . This paper is part of a general programme of treating explicit substitutions as the primary calculi from the point of view of foundations as well as applications. Here we investigate the property of strong normalization. To date all the proofs of strong normalization of typed calculi of explicit ..."
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. This paper is part of a general programme of treating explicit substitutions as the primary calculi from the point of view of foundations as well as applications. Here we investigate the property of strong normalization. To date all the proofs of strong normalization of typed calculi of explicit substitutions use a reduction to the strong normalization of classical calculus via the socalled \preservation of strong normalization" property. This paper develops a new approach, namely a direct proof that the strongly normalizing terms are precisely those typable under the intersectiontypes discipline. We also dene an eective perpetual strategy for the general calculus, give an inductive denition of the strongly normalizing terms, and furthermore show that normalization properties are essentially unaected by the inclusion of a rule for garbage collection. A key role is played by a certain general combinatorial lemma relating the reduction properties of two interacting abstract r...
Type and FlowDirected Compilation for Specialized Data Representations
, 2002
"... The combination of intersection and union types with ow types gives the compiler writer unprecedented exibility in choosing data representations in the context of a typed intermediate language. We present the design of such a language and the design of a framework for exploiting the type system to s ..."
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The combination of intersection and union types with ow types gives the compiler writer unprecedented exibility in choosing data representations in the context of a typed intermediate language. We present the design of such a language and the design of a framework for exploiting the type system to support multiple representations of the same data type in a single program. The framework can transform the input term, in a typesafe way, so that dierent data representations can be used in the transformed term  even if they share a use site in the pretransformed term. We have implemented a compiler using the typed intermediate language and instantiated the framework to allow specialized function representations. We test the compiler on a set of benchmarks and show that the compiletime performance is reasonable. We further show that the compiled code does indeed bene t from specialized function representations.