F-ing modules

Abstract

ML 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 F ω (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 F ω .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 with second-class modules and generative functors. We then extend this sublanguage with the ability to package modules as first-class values (a very simple extension, as it turns out) and OCaml-style applicative functors (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 for pure versus impure functors. 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.

The Essence of Dependent Object Types

“…MixML (Dreyer and Rossberg, 2008) drops the stratification requirement between module and term language and enables modules as first class values as well as mixin-style recursive definitions. More recent work models key aspects of modules as extensions of System F (Montagu and Rémy, 2009;Rossberg et al, 2014). The latest development, 1ML (Rossberg, 2015), unifies the ML module and core languages through an elaboration to System F ω .…”

Type soundness for dependent object types (DOT)

“…ML Module Systems 1ML [37] unifies the ML module and core languages through an elaboration to System F ω based on earlier such work [38]. Compared to DOT, the formalism treats recursive modules in a less general way and it only models fully abstract vs fully concrete types, not bounded abstract types.…”

Mechanizing Metatheory Without Typing Contexts

“…All these approaches turn out to be equivalent, but the cofinite quantification is gaining popularity in the programming language community because it eliminates typical renaming lemmas (but not all renaming lemmas). Recent work by Garrigue [6], Montagu [15], and Rossberg et al [19] reports the strength of cofinite quantification as well as its pitfalls.…”