We demonstrate the pragmatic value of the principal typing property, a property more general than ML's principal type property, by studying a type system with principal typings. The type system is based on rank 2 intersection types and is closely related to ML. Its principal typing property provides elegant support for separate compilation, including "smartest recompilation" and incremental type inference, and for accurate type error messages. Moreover, it motivates a novel rule for typing recursive definitions that can type many examples of polymorphic recursion.

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 DEXPTIME-complete. 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.

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We propose a rank 2 intersection type system for a language of modules built on a core ML-like language. The principal typing property of the rank 2 intersection type system for the core language plays a crucial role in the design of the type system for the module language. We first consider a "plain" notion of module, where a module is just a set of mutually recursive top-level definitions, and illustrate the notions of: module intrachecking (each module is typechecked in isolation and its interface, which is the set of typings of the defined identifiers, is inferred); interface interchecking (when linking modules, typechecking is done just by looking at the interfaces); interface specialization (interface intrachecking may require to specialize the typing listed in the interfaces); principal interfaces (the principal typing property for the type system of modules); and separate typechecking (looking at the code of the modules does not provide more type information than looking at their interfaces). Then we illustrate some limitations of the "plain" framework and extend the module language and the type system in order to overcome these limitations. The decidability of the system is shown by providing algorithms for the fundamental operations involved in module intrachecking and interface interchecking.

Let # be a rank-2 intersection type system. We say that a term is #-simple (or just simple when the system # is clear from the context) if system # can prove that it has a simple type. In this paper we propose new typing rules and algorithms that are able to type recursive definitions that are not simple. At the best of our knowledge, previous algorithms for typing recursive definitions in the presence of rank-2 intersection types allow only simple recursive definitions to be typed. The proposed rules are also able to type interesting examples of polymorphic recursion (i.e., recursive definitions rec {x = e} where di#erent occurrences of x in e are used with di#erent types). Moreover, the underlying techniques do not depend on particulars of rank-2 intersection, so they can be applied to other type systems.

We propose a rank 2 intersection type system for a language of modules built on a core ML-like language. The principal typing property of the rank 2 intersection type system for the core language plays a crucial role in the design of the type system for the module language. We first consider a "plain" notion of module, where a module is just a set of mutually recursive top-level definitions, and illustrate the notions of: module intrachecking (each module is typechecked in isolation and its interface, which is the set of typings of the defined identifiers, is inferred); interface interchecking (when linking modules, typechecking is done just by looking at the interfaces); interface specialization (interface intrachecking may require to specialize the typing listed in the interfaces); principal interfaces (the principal typing property for the type system of modules); and separate typechecking (looking at the code of the modules does not provide more type information than looking at their interfaces). Then we illustrate some limitations of the "plain" framework and extend the module language and the type system in order to overcome these limitations. The decidability of the system is shown by providing algorithms for the fundamental operations involved in module intrachecking and interface interchecking.