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Type systems based on intersection types have been studied extensively in recent years, both as tools for the analysis of the pure -calculus and, more recently, as the basis for practical programming languages. The dual notion, union types, also appears to have practical interest. For example, by refining types ordinarily considered as atomic, union types allow a restricted form of abstract interpretation to be performed during typechecking. The addition of second-order polymorphic types further increases the power of the type system, allowing interesting variants of many common datatypes to be encoded in the "pure" fragment with no type or term constants. This report summarizes a preliminary investigation of the expressiveness of a programming language combining intersection types, union types, and polymorphism.

this paper (Compagnoni, Intersection Types and Bounded Polymorphism 3 1994; Compagnoni, 1995) has been used in a type-theoretic model of object-oriented multiple inheritance (Compagnoni & Pierce, 1996). Related calculi combining restricted forms of intersection types with higher-order 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 proof-theoretic 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

Conditions on type preorders are provided in order to characterize the induced filter models for the #-calculus and some of its restrictions. Besides, two examples are given of filter models in which not all the continuous functions are representable. 1.