We present a new approach to inferring types in untyped object-oriented programs with inheritance, assignments, and late binding. It guarantees that all messages are understood, annotates the program with type information, allows polymorphic methods, and can be used as the basis of an op-timizing compiler. Types are finite sets of classes and subtyping is set inclusion. Using a trace graph, our algorithm constructs a set of conditional type constraints and computes the least solution by least fixed-point derivation.

We study an extension of the Hindley-Milner system with explicit type scheme annotations and type declarations. The system can express polymorphic function arguments, user-defined data types with abstract components, and structure types with polymorphic fields. More generally, all programs of the polymorphic lambda calculus can be encoded by a translation between typing derivations. We show that type reconstruction in this system can be reduced to the decidable problem of first-order unification under a mixed prefix.

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.

The Blue Calculus is a direct extension of both the lambda and the pi calculi. In a preliminary work from Gérard Boudol, a simple type system was given that incorporates Curry's type inference for the lambda-calculus. In the present paper we study an implicit polymorphic type system, adapted from the ML typing discipline. Our typing system enjoys subject reduction and principal type properties and we give results on the complexity for the type inference problem. These are interesting results for the blue calculus as a programming notation for higher-order concurrency.

DataTypes : : : : : : : : : : : : : : : : : : : : : : : : : 40 4.2 PolymorphicDataTypes : : : : : : : : : : : : : : : : : : : : : : : 40 vi 4.3 Existential DataTypes : : : : : : : : : : : : : : : : : : : : : : : : 41 4.4 DynamicTypes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 43 4.5 Dynamic Semantics : : : : : : : : : : : : : : : : : : : : : : : : : : 45 5 A Tiny Programming Language 48 5.1 Tiny Layout Conventions : : : : : : : : : : : : : : : : : : : : : : 48 5.2 Existential Type Variables : : : : : : : : : : : : : : : : : : : : : : 50 5.3 Type Signatures withHoles : : : : : : : : : : : : : : : : : : : : : 50 5.4 PolymorphicAbstractions : : : : : : : : : : : : : : : : : : : : : : 51 5.5 Top-Level Function Signatures : : : : : : : : : : : : : : : : : : : : 53 5.6 AlgebraicDataTypes : : : : : : : : : : : : : : : : : : : : : : : : 54 5.7 Restricted Type Synonyms and Newtype : : : : : : : : : : : : : : 55 5.8 Summary of Type System : : : : : : : : : : : : : : : : : : : : : : 56 5.9 DynamicTypes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 57 6 Systems Programming with Dynamic Types 59 6.1 Error Handling : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59 6.2 Inter-Process Communication : : : : : : : : : : : : : : : : : : : : 62 6.3 A Simple Address Server : : : : : : : : : : : : : : : : : : : : : : : 64 6.4 A Simple File Server : : : : : : : : : : : : : : : : : : : : : : : : : 66 6.5 A Capability System : : : : : : : : : : : : : : : : : : : : : : : : : 69 6.6 Distributed Inter-Process Communication : : : : : : : : : : : : : 75 6.7 A Compiler Interface : : : : : : : : : : : : : : : : : : : : : : : : : 78 6.8 AFunctional Compiler Interface : : : : : : : : : : : : : : : : : : 81 7 Dynamic Overloading 83 7.1 Dynamically Resolved Overloading : : ...

In this lecture we will present a tiny functional language and gradually enrich its type system. We shall cover the basic Curry-Hindley system and Wand's constraint-based algorithm for monomorphic type inference; brie y observe the Curry-Howard isomorphism and notice that logical formalism may serve as the inspiration for new type rules; present the polymorphic Milner system and the Damas-Milner algorithm for polymorphic type inference; see the Milner-Mycroft system for polymorphic recursion; and sketch the development of higher type systems. We will touch upon the relationship between types and logic and show how rules from logic may give inspiration for new type rules. En route we shall encounter the curious discovery that two algorithmic problems for type systems, which have been implemented in popular programming languages, have turned out to be respectively complete for exponential time and undecidable. 1 1