Encoding Types in ML-like Languages (Preliminary Version)

A Hindley-Milner type system such as ML's seems to prohibit typeindexed values, i.e., functions that map a family of types to a family of values. Such functions generally perform case analysis on the input types and return values of possibly different types. The goal of our work is to demonstrate how to program with type-indexed values within a Hindley-Milner type system.Our first approach is to interpret an input type as its correspondingvalue, recursively. This solution is type-safe, in the sense that the ML type system statically prevents any mismatch between the input type and function arguments that depend on this type.Such specific type interpretations, however, prevent us from combining different type-indexed values that share the same type. To meet this objection, we focus on finding a value-independent type encoding that can be shared by different functions. We propose and compare two solutions. One requires first-class and higher-order polymorphism, and, thus, is not implementable in the core language of ML, but itcan be programmed using higher-order functors in Standard ML of New Jersey. Its usage, however, is clumsy. The other approach uses embedding/projection functions. It appears to be more practical. We demonstrate the usefulness of type-indexed values through examples including type-directed partial evaluation, C printf-like formatting, and subtype coercions. Finally, we discuss the tradeoffs between our approach and some other solutions based on more expressive typing disciplines

Encoding types in ML-like languages

AbstractThis article presents several general approaches to programming with type-indexed families of values within a Hindley–Milner type system. A type-indexed family of values is a function that maps a family of types to a family of values. The function performs a case analysis on the input types and returns values of possibly different types. Such a case analysis on types seems to be prohibited by the Hindley–Milner type system. Our approaches solve the problem by using type encodings. The compile-time types of the type encodings reflect the types themselves, thereby making the approaches type-safe, in the sense that the underlying type system statically prevents any mismatch between the input type and the function arguments that depend on this type.A type encoding could be either value-dependent, meaning that the type encoding is tied to a specific type-indexed family, or value-independent, meaning that the type encoding can be shared by various type-indexed families. Our first approach is value-dependent: we simply interpret a type as its corresponding value. Our second approach provides value-independent type encodings through embedding and projection functions; they are universal type interpretations, in that they can be used to compute other type interpretations. We also present an alternative approach to value-independent type encodings, using higher-order functors.We demonstrate our techniques through applications such as C printf-like formatting, type-directed partial evaluation, and subtype coercions

Types For Modules

The programming language Standard ML is an amalgam of two, largely orthogonal, languages. The Core language expresses details of algorithms and data structures. The Modules language expresses the modular architecture of a software system. Both languages are statically typed, with their static and dynamic semantics specified by a formal definition. Over the past decade, Standard ML Modules has been the source of inspiration for much research into the type-theoretic foundations of modules languages. Despite these efforts, a proper type-theoretic understanding of its static semantics has remained elusive. In this thesis, we use Type Theory as a guideline to reformulate the unconventional static semantics of Modules, providing a basis for useful extensions to the Modules language. Our starting point is a stylised presentation of the existing static semantics of Modules, parameterised by an arbitrary Core language. We claim that the type-theoretic concepts underlying Modules are type parameterisation, type quantification and subtyping. We substantiate this claim by giving a provably equivalent semantics with an alternative, more type-theoretic presentation. In particular, we show that the notion of type generativity corresponds to existential quantification over types. In contrast to previous accounts, our analysis does not involve first-order dependent types. Our first extension generalises Modules to higher-order, allowing modules to take parameterised modules as arguments, and return them as results. We go beyond previous proposals for higher-order Modules by supporting a notion of type generativity. We give a sound and complete algorithm for type-checking higher-order Modules. Our second extension permits modules to be treated as first-class citizens of an ML-like Core language, greatly extending the range of computations on modules. Each extension arises from a natural generalisation of our type-theoretic semantics. This thesis also addresses two pragmatic concerns. First, we propose a simple approach to the separate compilation of Modules, which is adequate in practice but has theoretical limitations. We suggest a modified syntax and semantics that alleviates these limitations. Second, we study the type inference problem posed by uniting our extensions to higher-order and first-class modules with an implicitly-typed, ML-like Core language. We present a hybrid type inference algorithm that integrates the classical algorithm for ML with the type-checking algorithm for Modules

Principal Signatures for Higher-order Program Modules

In this paper we present a language for programming with higher-order modules.y The language, HML, is based on Standard ML in that it provides structures, signatures and functors. In HML, functors can be declared inside structures and specified inside signatures; this is not possible in Standard ML. We present an operational semantics for the static semantics of HML signature expressions with particular emphasis on the handling of sharing. As a justification for the semantics, we prove a theorem about the existence of principal signatures. This result is closely related to the existence of principal type schemes for functional programming languages with polymorphism. 1 Introduction Working on large programs involves manipulating large program units as well as working on the details of individual units. Such program units are sometimes called modules, especially if the programming language in question allows the programmer to name units and combine them in a controlled fashion. Since ..

Typed open programming : a higher-order, typed approach to dynamic modularity and distribution

In this dissertation we develop an approach for reconciling open programming the development of programs that support dynamic exchange of higher-order values with other processes with strong static typing in programming languages. We present the design of a concrete programming language, Alice ML, that consists of a conventional functional language extended with a set of orthogonal features like higher-order modules, dynamic type checking, higher-order serialisation, and concurrency. On top of these a flexible system of dynamic components and a simple but expressive notion of distribution is realised. The central concept in this design is the package, a first-class value embedding a module along with its interface type, which is dynamically checked whenever the module is extracted. Furthermore, we develop a formal model for abstract types that is not invalidated by the presence of primitives for dynamic type inspection, as is the case for the standard model based on existential quantification. For that purpose, we present an idealised language in form of an extended -calculus, which can express dynamic generation of types. This calculus is the first to combine and explore the interference of sealing and type inspection with higher-order singleton kinds, a feature for expressing sharing constraints on abstract types. A novel notion of abstracton kinds classifies abstract types. Higher-order type and kind coercions allow for modular translucent encapsulation of values at arbitrary type.In dieser Dissertation entwickeln wir einen programmiersprachlichen Ansatz zur Verbindung offener Programmierung der Entwicklung von Programmen, die das dynamische Laden und Austauschen höherstufiger Werte mit anderen Prozessen erlauben mit starker statischer Typisierung. Wir stellen das Design einer konkreten Programmiersprache namens Alice ML vor. Sie besteht aus einer konventionellen funktionalen Sprache, die um einen Satz orthogonaler Konzepte wie höherstufige Modularisierung, dynamische Typüberprüfung, höherstufige Serialisierung und Nebenläufigkeit erweitert wurde. Darauf aufbauend ist ein flexibles System dynamischer Komponenten sowie ein einfacher aber expressiver Ansatz fur Verteilung verwirklicht. Zentral ist dabei das Konzept eines Pakets (package), welches ein Modul in Kombination mit seinem Schnittstellentyp in einen Wert einbettet, und bei der Extraktion des Moduls eine dynamische Typüberprüfung vornimmt. Weiterhin entwickeln wir einen theoretischen Ansatz zur Modellierung von abstrakten Typen, welcher im Gegensatz zum herkömmlichen formalen Modell existentieller Quantifizierung auch in Gegenwart dynamischer Typinspektion gültig ist. Zu diesem Zweck definieren wir eine idealisierte Sprache in Form eines erweiterten λ-Kalküls, der dynamische Typgenerierung ausdrucken kann. Der Kalkül kombiniert diese erstmals mit höherstufigen Singleton Kinds, einem Sprachkonstrukt, welches Gleichheit von Typen ausdrücken kann. Zur Klassifizierung abstrakter Typen werden Abstraktions-Kinds als verwandtes Konzept entwickelt. Höherstufige Konversionen auf Term- und Typebene erlauben zudem die nachträgliche modulare Enkapsulierung von Werten beliebigen Typs