A partial type inference technique should come with a simple and precise specification, so that users predict its behavior and understand the error messages it produces. Local type inference techniques attain this simplicity by inferring missing type information only from the types of adjacent syntax nodes, without using global mechanisms such as unification variables. The paper reports on our experience with programming in a full-featured programming language including higher-order polymorphism, subtyping, parametric datatypes, and local type inference. On the positive side, our experiments on several nontrivial examples confirm previous hopes for the practicality of the type inference method. On the negative side, some proposed extensions mitigating known expressiveness problems turn out to be unsatisfactory on close examination. 1 Introduction It is widely believed that a polymorphic programming language should provide some form of type inference, to avoid discouraging programming ...

We present the first algorithm for reconstructing the types and effects of expressions in the presence of first class procedures in a polymorphic typed language. Effects are static descriptions of the dynamic behavior of expressions. Just as a type describes what an expression computes, an effect describes how an expression computes. Types are more complicated to reconstruct in the presence of effects because the algebra of effects induces complex constraints on both effects and types. In this paper we show how to perform reconstruction in the presence of such constraints with a new algorithm called algebraic reconstruction, prove that it is sound and complete, and discuss its practical import. This research was supported by DARPA under ONR Contract N00014-89-J-1988. 1

We show that the problem of partial type inference in the nthb-order polymorphic X-calculus is equivalent to nth-order unification. On the one hand, this means that partial type inference in polymorphic X-calculi of order 2 or higher is undecidable. On the other hand, higher-order unification is often tractable in practice, and our translation entails a very useful algorithm for partial type inference in the w-order polymorphic X-calculus. We present an implementation in AProlog in full.

Girard and Reynolds independently invented System F (a.k.a. the second-order polymorphically typed lambda calculus) to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking . Typability asks for a term whether there exists some type it can be given. Type checking asks, for a particular term and type, whether the term can be given that type. The decidability of these problems has been settled for restrictions and extensions of F and related systems and complexity lower-bounds have been determined for typability in F, but this report is the rst to resolve whether these problems are decidable for System F. This report proves that type checking in F is undecidable, by a reduction from semiuni cation, and that typability in F is undecidable, by a reduction from type checking. Because there is an easy reduction from typability to typ...

Languages like ML and Haskell encourage the view of values as first-class entities that can be passed as arguments or results of functions, or stored as components of data structures. The same languages o#er parametric polymorphism, which allows the use of values that behave uniformly over a range of di#erent types. But the combination of these features is not supported--- polymorphic values are not first-class. This restriction is sometimes attributed to the dependence of such languages on type inference, in contrast to more expressive, explicitly typed languages, like System F, that do support first-class polymorphism. This paper uses relationships between types and logic to develop a type system, FCP, that supports first-class polymorphism, type inference, and also first-class abstract datatypes. The immediate result is a more expressive language, but there are also long term implications for language design. 1

We prove that partial type reconstruction for the pure polymorphic *-calculus is undecidable by a reduction from the second-order unification problem, extending a previous result by H.-J. Boehm. We show further that partial type reconstruction remains undecidable even in a very small predicative fragment of the polymorphic *-calculus, which implies undecidability of partial type reconstruction for * ML as introduced by Harper, Mitchell, and Moggi.

... unification problems. Then, in this framework, we develop a new unification algorithm for a-calculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (object-languages). The rich structure of a typed-calculus,asopposedtotraditional,rst- generalideaistousea-calculusasameta-languageforrepresentingvariousotherlanguages thelattercase,thealgorithmisincomplete,thoughstillquiteusefulinpractice. Thelastpartofthedissertationprovidesexamplesoftheusefulnessofthealgorithms.The algorithmrstfordependentproduct()types,andsecondforimplicitpolymorphism.In involvessignicantcomplicationsnotarisingHuet'scorrespondingalgorithmforthesimply orderabstractsyntaxtrees,allowsustoexpressrules,e.g.,programtransformationand typed-calculus,primarilybecauseitmustdealwithill-typedterms.Wethenextendthis Wecanthenuseunicationinthemeta-languagetomechanizeapplicationoftheserules.

In the last ten years declaration-free programming languages with a polymorphic typing discipline (ML, B) have been developed to approximate the flexibility and conciseness of dynamically typed languages (LISP, SETL) while retaining the safety and execution efficiency of conventional statically typed languages (Algol68, Pascal). These polymorphic languages can be type checked at compile time, yet allow functions whose arguments range over a variety of types. We investigate several polymorphic type systems, the most powerful of which, termed Milner-Mycroft Calculus, extends the so-called let-polymorphism found in, e.g., ML with a polymorphic typing rule for recursive definitions. We show that semi-unification, the problem of solving inequalities over firstorder terms, characterizes type checking in the Milner-Mycroft Calculus to polynomial time, even in the restricted case where nested definitions are disallowed. This permits us to extend some infeasibility results for related combinato...

We propose a type system ML F that generalizes ML with first-class polymorphism as in System F. We perform partial type reconstruction. As in ML and in opposition to System F, each typable expression admits a principal type, which can be inferred. Furthermore, all expressions of ML are well-typed, with a possibly more general type than in ML, without any need for type annotation. Only arguments of functions that are used polymorphically must be annotated, which allows to type all expressions of System F as well.

We present the first type reconstruction system which combines the implicit typing of ML with the full power of the explicitly typed second-order polymorphic lambda calculus. The system will accept ML-style programs, explicitly typed programs, and programs that use explicit types for all first-class polymorphic values. We accomplish this flexibility by providing both generic and explicitly-quantified polymorphic types, as well as operators which convert between these two forms of polymorphism. This type reconstruction system is an integral part of the FX-89 programming language. We present a type reconstruction algorithm for the system. The type reconstruction algorithm is proven sound and complete with respect to the formal typing rules.