Posted on January 25, 2017 by Troels Henriksen

When most programmers think of module systems, they think of rather utilitarian systems for namespace control and splitting programs across multiple files. And in most languages, the module system is indeed little more than this. But when it became time to add a module system to Futhark, we eyed an opportunity to address several long-standing weaknesses in one fell swoop. We observed that an ML-style module system would allow us to add modularity, namespace control, polymorphism, and higher-order functions, or at least functional equivalents. In short, we would gain the ability to write generic code in Futhark; something we have sorely missed. And crucially: the ML module system is defined entirely in terms of compile-time substitution. This gives us a guarantee that no matter which abstractions and parametric behaviour is encoded by the programmer via the module system, we can strip it away immediately, and produce a program in the monomorphic first-order core language that is expected by our optimising compiler. This is a core part our of language design philosophy.

This blog post will give an introduction to the Futhark module system. Prior knowledge of ML-style module systems is not required. If you do already have ML experience, note that I will be using Futhark’s syntax, which differs somewhat from both Standard ML and OCaml. Most notably, Futhark makes use of curly braces; an inescapable prerequisite for any language that hopes for adoption.

At the most basic level, a module (called a struct in Standard ML) is merely a collection of declarations:

module AddI32 = { type t = i32 let add (x: t) (y: t): t = x + y let zero: t = 0 }

Now, AddI32.t is an alias for the type i32 , and AddI32.add is a function that adds two values of type i32 . The only peculiar thing about this notation is the equal sign before the opening brace. The declaration above is actually a combination of a module binding:

module ADDI32 = ...

And a module expression:

{ type t = i32 let add (x: t) (y: t): t = x + y let zero: t = 0 }

In this case, the module expression is just some declarations enclosed in curly braces. But, as the name suggests, a module expression is just some expression that returns a module. A module expression is syntactically and conceptually distinct from a regular value expression, but serves much the same purpose. The module language is designed such that evaluating a module expression can always be done at compile time.

Apart from a sequence of declarations, a module expression can also be merely the name of another module:

module Foo = AddInt32

Now every name defined in AddInt32 is also available in Foo . At compile-time, only a single version of the add function is defined.

What we have seen so far is nothing more than a simple namespacing mechanism. The ML module system only becomes truly powerful once we introduce module types and parametric modules (in Standard ML, these are called signatures and functors).

A module type is the counterpart to a value type. It describes which names are defined, and as what. We can define a module type that describes AddInt32 :

module type Int32Adder = { type t = i32 val add: t -> t -> t val zero: t }

As with modules, we have the notion of a module type expression. In this case, the module type expression is a sequence of specs enclosed in curly braces. A spec is a requirement of how some name must be defined: as a value (including functions) of some type, as a type abbreviation, or as an abstract type (which we will return to later).

We can assert that some module implements a specific module type via module type ascription:

module Foo = AddInt32 : Int32Adder

Syntactical sugar that allows us to move the module type to the left of the equal sign makes a common case look smoother:

module AddInt32: Int32Adder = { ... }

When we are ascribing a module with a module type, the module type functions as a filter, removing anything not explicitly mentioned in the module type:

module Bar = AddInt32 : { type t = int val zero: t }

An attempt to access Bar.add will result in a compilation error, as the ascription has hidden it. This is known as an opaque ascription, because it obscures anything not explicitly mentioned in the module type. The module systems in Standard ML and OCaml support both opaque and transparent ascription, but in Futhark we support only the former. This example also demonstrates the use of an anonymous module type. Module types work much like structural types known from e.g. Go (“compile-time duck typing”), and are named only for convenience.

We can use type ascription with abstract types to hide the definition of a type from the users of a module:

module Speeds: { type thing val car: thing val plane: thing val futhark: thing val speed: thing -> int } = { type thing = int let car: thing = 0 let plane: thing = 1 let futhark: thing = 2 let speed (x: thing): int = if x == car then 120 else if x == plane then 800 else if x == futhark then 10000 else 0 -- will never happen }

The (anonymous) module type asserts that a distinct type thing must exist, but does not mention its definition. There is no way for a user of the Speeds module to do anything with a value of type Speeds.thing apart from passing it to Speeds.speed (except putting it in an array or tuple, or returning it from a function). Its definition is entirely abstract. Furthermore, no values of type Speeds.thing exist except those that are created by the Speeds module.

While module types serve some purpose for namespace control and abstraction, their most interesting use is in the definition of parametric modules. A parametric module is conceptually equivalent to a function. Where a function takes a value as input and produces a value, a parametric module takes a module and produces a module. For example, given a module type:

module type Monoid = { type t val add: t -> t -> t val zero: t }

We can define a parametric module that accepts a module satisfying the Monoid module type, and produces a module containing a function for collapsing an array:

module Sum(M: Monoid) = { let sum (a: []M.t): M.t = reduce M.add M.zero a }

There is an implied assumption here, which is not captured by the type system: the function add must be associative and have zero as its neutral element. These constraints are from the parallel semantics of reduce , and the algebraic concept of a monoid. Note that in Monoid , no definition is given of the type t - we only assert that there must be some type t , and that certain operations are defined for it.

We can use the parametric module Sum thus:

module SumI32s = Sum(AddInt32)

We can now refer to the function SumI32s.sum , which has type []i32 -> i32 . The type is only abstract inside the definition of the parametric module. We can instantiate Sum again with another module; this one anonymous:

module Prod64s = Sum({ type t = f64 let add (x: f64) (y: f64): f64 = x * y let zero: f64 = 1.0 })

The function Prodf64s.sum has type []f64 -> f64 , and computes the product of an array of numbers (we should probably have picked a more generic name than sum for this function).

Operationally, each application of a parametric module results in its definition being duplicated and references to the module parameter replace by references to the concrete module argument. This is quite similar to how C++ templates are implemented. Indeed, parametric modules can be seen as a simplified variant with no specialisation, and with module types to ensure rigid type checking. In C++, a template is type-checked when it is instantiated, whereas a parametric module is type-checked when it is defined.

Parametric modules, like other modules, can contain more than one declaration. This is useful for giving related functionality a common abstraction, for example to implement linear algebra operations that are polymorphic over the type of scalars. This example uses an anonymous module type for the module parameter, and the open declaration, which brings the names from a module into the current scope:

module Linalg(M: { type scalar val zero: scalar val add: scalar -> scalar -> scalar val mul: scalar -> scalar -> scalar }) = { open M let dotprod (xs: [n]scalar) (ys: [n]scalar): scalar = reduce add zero (zipWith mul xs ys) let matmul (xss: [n][p]scalar) (yss: [p][m]scalar): [n][m]scalar = map (\xs -> map (dotprod xs) (transpose yss)) xss }

We are using these facilities to carve a Futhark standard library, although it is still very sparse.

The above examples of parametric modules could equally well have been implemented using polymorphic higher-order functions. Indeed, there has been work in the ML community on blurring the phase distinction between modules and values. A module can be viewed as nothing but a record containing types and values. However, for Futhark, we like the phase distinction. We want to be sure we can compile away all the higher-order behaviour, in order to guarantee simple lightweight code that does not have to keep function pointers or closure objects around. An ML-style module system gives us just this, and is very simple to implement. The implementation in Futhark was added in just one week, at a cost of less than 500 lines of code, although it did build on an earlier embryonic module system without module types or parametric modules.

Of course, we are well aware that the module system is significantly more verbose and clunky than proper higher-order functions. We intend to add shorthand forms for just those cases that can be encoded in the module system, likely including some mechanism similar to type classes in order to permit ad-hoc bounded polymorphism. With our module system, we now have a basis that is powerful enough to encode the generic code we need, as well as a compilation model that is able to remove all the overhead of abstraction. All we have left to do is add more convenient interfaces to the sound core functionality.