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Programming with Intersection Types and Bounded Polymorphism
, 1991
"... representing the official policies, either expressed or implied, of the U.S. Government. ..."
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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 polymorphism ..."
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Cited by 37 (0 self)
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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
Polarized HigherOrder Subtyping
, 1997
"... The calculus of higher order subtyping, known as F ω ≤ , a higherorder polymorphic λcalculus with subtyping, is expressive enough to serve as core calculus for typed objectoriented languages. The versions considered in the literature usually support only pointwise subtyping of type operators, whe ..."
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The calculus of higher order subtyping, known as F ω ≤ , a higherorder polymorphic λcalculus with subtyping, is expressive enough to serve as core calculus for typed objectoriented languages. The versions considered in the literature usually support only pointwise subtyping of type operators, where two types S U and T U are in subtype relation, if S and T are. In the widely cited, unpublished note [Car90], Cardelli presents F ω ≤ in a more general form going beyond pointwise subtyping of type applications in distinguishing between monotone and antimonotone operators. Thus, for instance, T U1 is a subtype of T U2, if U1 ≤ U2 and T is a monotone operator. My thesis extends F ω ≤ by polarized application, it explores its proof theory, establishing decidability of polarized F ω ≤. The inclusion of polarized application rules leads to an interdependence of the subtyping and the kinding system. This contrasts with pure F ω ≤ , where subtyping depends on kinding but not vice versa. To retain decidability of the system, the equalbounds subtyping rule for alltypes is rephrased in the polarized setting as a mutualsubtype requirement of the upper bounds.
Parametricity as Subtyping
, 1992
"... A polymorphic function is parametric if it has uniform behavior for all type parameters. This property is useful when writing, reasoning about, and compiling functional programs. We show how to syntactically define and reason about parametricity in a language with intersection types and bounded poly ..."
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A polymorphic function is parametric if it has uniform behavior for all type parameters. This property is useful when writing, reasoning about, and compiling functional programs. We show how to syntactically define and reason about parametricity in a language with intersection types and bounded polymorphism. Within this framework, parametricity is subtyping, and reasoning about parametricity becomes reasoning about the welltypedness of terms. This work also demonstrates the expressiveness of languages that combine intersection types and bounded polymorphism. 1 Introduction A polymorphic function is parametric if it uses the same algorithm regardless of which type parameter is instantiated. As a consequence, it has a uniform behavior over all type parameters, in the sense that for any related inputs, the function produces related outputs. Let us look at an example. Consider the polymorphic "doubling" function double = ø : f ø!ø : x ø : f (fx) : 8ø: (ø!ø )!(ø!ø ) : By passing int and...
AntiSymmetry of HigherOrder Subtyping
 In Proceedings of the 8th Annual Conference on Computer Science Logic (CSL’99), J. Flum and M. RodríguezArtalejo, Eds. SpringerVerlag LNCS 1683
, 1999
"... . This paper shows that the subtyping relation of a higherorder lambda calculus, F ! , is antisymmetric. It exhibits the rst such proof, establishing in the process that the subtyping relation is a partial orderreexive, transitive, and antisymmetric up to equality. While a subtyping relat ..."
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. This paper shows that the subtyping relation of a higherorder lambda calculus, F ! , is antisymmetric. It exhibits the rst such proof, establishing in the process that the subtyping relation is a partial orderreexive, transitive, and antisymmetric up to equality. While a subtyping relation is reexive and transitive by denition, antisymmetry is a derived property. The result, which may seem obvious to the nonexpert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping oers a powerful new technology to solve the problem: of particular importance is our extended rule for the wellformedness of types with head variables. The paper also gives a presentation of F ! without a relation for equality, apparently the rst such, and shows its equivalence with the traditional presentation. 1 Introduction Objectoriented programming languages such as Smalltalk, C++, Modula 3, and ...