System Fω Research Articles

Lambda: the ultimate sublanguage (experience report)

We describe our experience teaching an advanced typed functional programming course based around the use of System Fω as a programming language.

1ML – Core and modules united

AbstractML is two languages in one: there is thecore, with types and expressions, and there aremodules, with signatures, structures, and functors. Modules form a separate, higher-order functional language on top of the core. There are both practical and technical reasons for this stratification; yet, it creates substantial duplication in syntax and semantics, and it imposes seemingly unnecessary limits on expressiveness because it makes modules second-class citizens of the language. For example, selecting one among several possible modules implementing a given interface cannot be made a dynamic decision. Language extensions allowing modules to be packaged up as first-class values have been proposed and implemented in different variations. However, they remedy expressiveness only to some extent and tend to be even more syntactically heavyweight than using second-class modules alone. We propose a redesign of ML in which modules are truly first-class values, and core and module layers are unified into one language. In this “1ML”, functions, functors, and even type constructors are one and the same construct; likewise, no distinction is needed between structures, records, or tuples. Or viewed the other way round, everything is just (“a mode of use of”) modules. Yet, 1ML does not require dependent types: its type structure is expressible in terms of plain System Fω, with a minor variation of ourF-ing modulesapproach. We introduce both an explicitly typed version of 1ML and an extension with Damas–Milner-style implicit quantification. Type inference for this language is not complete, but, we argue, not substantially worse than for Standard ML.

1ML – core and modules united (F-ing first-class modules)

ML is two languages in one: there is the core, with types and expressions, and there are modules, with signatures, structures and functors. Modules form a separate, higher-order functional language on top of the core. There are both practical and technical reasons for this stratification; yet, it creates substantial duplication in syntax and semantics, and it reduces expressiveness. For example, selecting a module cannot be made a dynamic decision. Language extensions allowing modules to be packaged up as first-class values have been proposed and implemented in different variations. However, they remedy expressiveness only to some extent, are syntactically cumbersome, and do not alleviate redundancy. We propose a redesign of ML in which modules are truly first-class values, and core and module layer are unified into one language. In this "1ML", functions, functors, and even type constructors are one and the same construct; likewise, no distinction is made between structures, records, or tuples. Or viewed the other way round, everything is just ("a mode of use of") modules. Yet, 1ML does not require dependent types, and its type structure is expressible in terms of plain System Fω, in a minor variation of our F-ing modules approach. We introduce both an explicitly typed version of 1ML, and an extension with Damas/Milner-style implicit quantification. Type inference for this language is not complete, but, we argue, not substantially worse than for Standard ML. An alternative view is that 1ML is a user-friendly surface syntax for System Fω that allows combining term and type abstraction in a more compositional manner than the bare calculus.

F-ing modules

AbstractML modules are a powerful language mechanism for decomposing programs into reusable components. Unfortunately, they also have a reputation for being “complex” and requiring fancy type theory that is mostly opaque to non-experts. While this reputation is certainly understandable, given the many non-standard methodologies that have been developed in the process of studying modules, we aim here to demonstrate that it is undeserved. To do so, we present a novel formalization of ML modules, which defines their semantics directly by a compositional “elaboration” translation into plain System Fω(the higher-order polymorphic λ-calculus). To demonstrate the scalability of our “F-ing” semantics, we use it to define a representative, higher-order ML-style module language, encompassing all the major features of existing ML module dialects (except for recursive modules). We thereby show that ML modules are merely a particular mode of use of System Fω.To streamline the exposition, we present the semantics of our module language in stages. We begin by defining a subset of the language supporting a Standard ML-like language withsecond-classmodules andgenerativefunctors. We then extend this sublanguage with the ability to package modules asfirst-classvalues (a very simple extension, as it turns out) and OCaml-styleapplicativefunctors (somewhat harder). Unlike previous work combining both generative and applicative functors, we do not require two distinct forms of functor or signature sealing. Instead, whether a functor is applicative or not depends only on the computational purity of its body. In fact, we argue that applicative/generative is rather incidental terminology forpureversusimpurefunctors. This approach results in a semantics that we feel is simpler and more natural than previous accounts, and moreover prohibits breaches of abstraction safety that were possible under them.

A framework for extensible languages

Extensible programming languages such as SugarJ or Racket enable programmers to introduce customary language features as extensions of the base language. Traditionally, systems that support language extensions are either (i) agnostic to the base language or (ii) only support a single base language. In this paper, we present a framework for language extensibility that turns a non-extensible language into an extensible language featuring library-based extensible syntax, extensible static analyses, and extensible editor support. To make a language extensible, our framework only requires knowledge of the base language's grammar, the syntax for import statements (which activate extensions), and how to compile base-language programs. We have evaluated the generality of our framework by instantiating it for Java, Haskell, Prolog, JavaScript, and System F ω , and by studying existing module-system features and their support in our framework.

Strong normalisation in two Pure Pattern Type Systems

Pure Pattern Type Systems (P2TS) combine the frameworks and capabilities of rewriting and λ-calculus within a unified setting. Their type systems, which are adapted from Barendregt's λ-cube, are especially interesting from a logical point of view. Until now, strong normalisation, which is an essential property for logical soundness, has only been conjectured: in this paper, we give a positive answer for the simply-typed system and the dependently-typed system.The proof is based on a translation of terms and types fromP2TSinto the λ-calculus. First, we deal with untyped terms, ensuring that reductions are faithfully mimicked in the λ-calculus. For this, we rely on an original encoding of the pattern matching capability ofP2TSinto the System Fω.Then we show how to translate types: the expressive power of System Fω is needed in order to fully reproduce the original typing judgments ofP2TS. We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalisation of simply-typedP2TSterms. The strong normalisation with dependent types is in turn obtained by an intermediate translation into simply-typed terms.

Operational interpretations of an extension of Fω with control operators

AbstractWe study the operational semantics of an extension of Girard's System Fω with two control operators: an abort operation that abandons the current control context, and a callcc operation that captures the current control context. Two classes of operational semantics are considered, each with a call-by-value and a call-by-name variant, differing in their treatment of polymorphic abstraction and instantiation. Under the standard semantics, polymorphic abstractions are values and polymorphic instantiation is a significant computation step; under the ML-like semantics evaluation proceeds beneath polymorphic abstractions and polymorphic instantiation is computationally insignificant. Compositional, type-preserving continuation-passing style (cps) transformation algorithms are given for the standard semantics, resulting in terms on which all four evaluation strategies coincide. This has as a corollary the soundness and termination of well-typed programs under the standard evaluation strategies. In contrast, such results are obtained for the call-by-value ML-like strategy only for a restricted sub-language in which constructor abstractions are limited to values. The ML-like call-by-name semantics is indistinguishable from the standard call-by-name semantics when attention is limited to complete programs.

Type Inference: Some Results, Some Problems1

In this paper we investigate the type inference problem for a large class of type assignment systems for the λ-calculus. This is the problem of determining if a term has a type in a given system. We discuss, in particular, a collection of type assignment systems which correspond to the typed systems of Barendregt’s “cube”. Type dependencies being shown redundant, we focus on the strongest of all, Fω, the type assignment version of the system Fω of Girard. In order to manipulate uniformly type inferences we give a syntax directed presentation of Fω and introduce the notions of scheme and of principal type scheme. Making essential use of them, we succeed in reducing the type inference problem for Fω to a restriction of the higher order semi-unification problem and in showing that the conditional type inference problem for Fω is undecidable. Throughout the paper we call attention to open problems and formulate some conjectures.