Posted on August 3, 2019 by Troels Henriksen

Futhark is not a large or particularly innovative language. We prefer to keep the novelties in the compiler, and only include language features that have already proven their worth in existing functional languages. One exception to this principle is size annotations, by which functions can express pre- and post-conditions on their arguments and return values. The classic example is matrix multiplication, where we can state that the innermost sizes of the array arguments must match, and that the shape of the result corresponds to the shape of the arguments:

let matmul [n][m][p] (x: [n][m]i32) (y: [m][p]i32): [n][p]i32 = matmul [n][m][p] (x: [n][m]i32) (y: [m][p]i32): [n][p]i32 = ...

The leading [n][m][p] are not ordinary parameters, but rather size parameters. They are not passed explicitly when calling matmul , but rather inferred from the matrices passed for x and y . Size parameters are in scope inside the function definition as i32 -typed variables, which means we can define a function for returning the length of an array as thus:

let length [n] 'a (xs: [n]a) = n length [n] 'a (xs: [n]a) = n

The length function (which is also polymorphic in the element type of xs ) does not use any of the values of xs - it just returns the size n .

We can even write functions where we state that the size of an array we return is the same as the value of one of our parameters, like in iota :

let iota (n: i32) : [n]i32 = 0 ..<n iota (n: i32) : [n]i32 =..

For simplicity, dimensions can be only names or constants, not compound expressions. I’ll return to the ramifications of this.

We have found size annotations to be very useful for understanding code, and it is one of the language features that Futhark programmers (few as they are) tend to single out as particularly useful, as size errors are some of the most common problems in array programming. Unfortunately, size annotations have one major flaw: they are not truly part of the type system. Currently, size annotations are ignored by the type checker and verified at run-time.

In this post, I will outline an idea for moving from size annotations to checkable size types. The design is not yet fully formed, as we shall see. While size types have a strong connection to dependent types, the way I intend to use them in Futhark are not quite the same as in truly dependently typed languages like Idris, so I cannot just take a design from the latest POPL paper. In particular, I want to preserve the Futhark current programming experience as much as possible, notably including preserving solid type inference. This is not so much because there is all that much Futhark code that I wish to avoid breaking, but because Futhark is currently a rather simple, accessible, and fun language, and I would like to keep it that way.

The idea behind size types is based on an extension of classic Hindley-Milner type inference. So first, let us review what that’s all about. There are two main concepts we must understand: instantiation and let-generalisation.

We will get to type inference shortly. For now, consider an explicitly typed identity function in Futhark:

let id 'a (x: a) : a = x id 'a (x: a) : a = x

This function is known to the type checker by the following type scheme:

val id 'a : a -> a id 'a : a -> a

The type scheme contains all the information that is needed to check uses of the function. Specifically, the type scheme tells us that id has a single type parameter ( a ). Consider now an application:

2 id

where i has type i32 . Note that an explicit argument is not provided for the type parameter. Whenever a reference to a polymorphic definition such as id is encountered by the type checker, its type scheme is instantiated. The instantiation procedure generates a fresh type variable for each type parameter, and then replaces each type parameter in the type with its corresponding type variable. Type variables are a bit of machinery that stand in for currently unknown types. A type variable can be unified with another type, which will replace all instances of the type variable with the other type.

For example, in the application id 2 , the id function might be instantiated by generating a fresh type variable t0 , and replacing a with t0 in the type scheme of id , such that the type of this instance of id now has type t0 -> t0 . Note the absence of type parameters: these are only present in type schemes, not types. (Higher-ranked types are different, but these do not exist in Futhark.) Since we are applying id to 2 , the type checker will unify the parameter type of id ( t0 ) with the type of 2 ( i32 ), producing the substitution t0 ⟼ i32 , which is then applied whenever t0 occurs. Since t0 also occurs in the return type of id , we find that the final type of id 2 is i32 .

Let-generalisation can be seen as the dual to instantiation, in that it turns un-unified type variables into type parameters in order to infer polymorphic definitions. The idea is simple. Whenever we have a let -binding

let x = e... x = e...

then if any type variables constructed while inferring the type of e remain in the type of e , those type variables are turned into type parameters when constructing the type scheme for x .

For example, consider the following definition:

let f x = x f x = x

When new names are bound (here, the parameter x ), we generate a new type variable, say tx . Hence, the body of f has type tx , and f as a whole has type tx -> tx , where tx is a type variable. Let-generalisation finds all such type variables and turn them into type parameters for the type scheme of f :

val f 'tx : tx -> tx f 'tx : tx -> tx

There need not be any relationship in naming between type parameters and type variables. In practice, type variables tend to have ugly internal names (because many of them are typically generated during type-checking nontrivial functions), while we would like inferred type parameters to have short and clean names. Thus, the compiler may perform some renaming before constructing the type scheme.

In practice, and in Futhark, let-generalisation is only done for let -bindings of functional type. We shall later see an example where this causes some problems.

The type variables discussed so far have been placeholders that could be replaced (through unification) with concrete types. We call such types nonrigid. In contrast, rigid types do not unify with anything but themselves. The most common source of rigid types are primitive types and explicit type parameters:

let g 'a (x: a) : i32 = x g 'a (x: a) : i32 = x

Here we are writing a function that tries to turn an a into an i32 . Clearly this should not be well typed. And indeed it is not: both a and i32 are rigid, so they cannot be unified with each other.

As a contrast, consider this contrived function:

let h 'a x : a = x h 'a x : a = x

Here the type checker will generate a nonrigid type variable xa for x , unify xa with a to produce the substitution xa ⟼ a , and finally infer the following type scheme:

let h 'a : a -> a h 'a : a -> a

Some presentations do not have the notion of rigid type variables at all - they simply consider them types, rather than type variables. However, for size types, it is convenient to treat them as a degenerate class of variables, as we shall see.

The basic idea is pretty simple: have size variables that act much as type variables, using the same unification machinery. We have a distinction between rigid sizes, which cannot unify with anything but themselves, and nonrigid size variables, which do unify. When unifying two array types [d1]t1 and [d2]t2 we unify d1 with d2 and t1 with t2 .

Consider the type of zip :

val zip [n] 'a 'b : [n]a -> [n]b -> [n](a,b) zip [n] 'a 'b : [n]a -> [n]b -> [n](a,b)

Now suppose that we are type-checking the following application:

zip xs ys

The type scheme of zip is instantiated and each type- and size parameter replaced with a new non-rigid type variable. Let us say that post-instantiation, this occurrence of zip has the following type:

When checking the application of zip to xs , we have to unify the parameter type [d0]t1 with the type of xs , which we will suppose is [10]bool . The unification succeeds and produces the following substitutions:

10 d0 ⟼ t1 ⟼ bool

This means that we infer the type of zip xs as [10]t2 -> [10](bool, t2) . When we then try to apply this to ys , unification will succeed only if the size of ys can be unified with the constant size 10 (so, ys must be an 10 element array).

Let-generalisation functions more or less the same. Suppose we are type-checking the following let -binding of a lambda abstraction:

let f = \xs ys -> zip xs ys f = \xs ys -> zip xs ys

After unification, the right-hand side will be inferred to have type [d0]t1 -> [d0]t2 -> [d0](t1, t2) , where d0 is a nonrigid size variable and t1 , and t2 are nonrigid type variables. Let-generalisation then takes place, turning the former into a size parameter, and the latter two into type parameters, yielding the following type scheme:

val f [d0] 't1 't2 = [d0]t1 -> [d0]t2 -> [d0](t1, t2) f [d0] 't1 't2 = [d0]t1 -> [d0]t2 -> [d0](t1, t2)

When replacing a type variable with an array type, the dimensions of that array type are propagated as well. This has the consequence that if the following fully polymorphic function is called with two arrays, those arrays must have the same size:

let pair 'a (x: a) (y: a) = (x, y) pair 'a (x: a) (y: a) = (x, y)

This restriction is actually already in place in Futhark (although by a different mechanism), and was added to slowly prepare the way for size types.

This simple mechanism is enough to handle a surprising majority of the cases that occur even in nontrivial Futhark programs. However, we also need to think about how to handle less well behaved programs.

Consider the filter function, which removes those elements of an array for which some function returns false:

val filter [n] 'a = (a -> bool) -> [n]a -> [?]a filter [n] 'a = (a -> bool) -> [n]a -> [?]a

What should we put in place of the question mark? The only non-constant size available to use is n , and that is clearly not correct. We call the size of such an array existential, based on the “∃” quantifier from logic: we know that the array must have some size, but we cannot know what it is until the function returns. To handle such cases, we just leave the dimension empty:

val filter [n] 'a = (a -> bool) -> [n]a -> []a filter [n] 'a = (a -> bool) -> [n]a -> []a

This is permitted only in declarations like type schemes and when annotating parameter types: when type-checking an expression, we would like to have the invariant that all array dimensions have a size, although that size may well be a size variable. To accomplish this, whenever we type-check an application of a function with empty dimensions in its return type, we instantiate those empty dimensions with new rigid size variables. The rigidity is crucial, as it means the shape of the array cannot match the shape of any other array. For example, in the expression

zip (filter p xs) ys

there is no expression ``ys`` that would make this well-typed. In most cases, when we want to do something interesting with an array of existential size, we will have to insert an explicit size coercion. These coercions are dynamically checked, and change the type of the array as far as the type checker is concerned:

10 ]bool) ys zip (filter p xs : []bool) ys

This convinces the type checker that filter p xs has type [10]bool . Only the sizes of each dimension may be changed this way - the rank and element type of the array must be as otherwise inferred. Size coercions are intended to be the sole syntactic construct that can fail at run-time with a shape error.

Similarly, we may have if expressions where the two branches return arrays of two different sizes:

if b then iota 10 iota else iota 20 iota

Those dimensions that do not match will have their size replaced with a rigid size variable.

Another case of rigid sizes occurs when a function with a size-dependent type is given an argument for its size parameter that cannot syntactically be a size. Consider again iota :

val iota : (n: i32) -> [n]i32 iota : (n: i32) -> [n]i32

The type of iota x should be [x]i32 , but what about iota (x+1) ? Syntactically, there is no such thing as a type [x+1]i32 . While it is likely we may lift this restriction some day (it is in place to avoid having to solve complicated arithmetic problems during unification), for now such an application will result in a rigid size. A simple workaround is to let -bind x+1 to a name and then passing that name to iota .

All of this is work-in-progress. While I have a branch implementing most of this (and it is able to compile most of our benchmarks and tests), there are still unsolved problems in both implementation and concept. As an example of the latter, consider the following program:

let f xs = zip (filter p xs) xs f xs = zip (filter p xs) xs

By the rules discussed so far, xs will initially be given the type [d0]t1 , where d0 is a nonrigid size variable, filter p xs will have type [d1]t1 where d1 is a rigid type variable, and then due to the zip we will produce the substitution d0 ⟼ d1 , and so the whole type of f will be [d1]t1 -> [d1](t1, t2) , where d1 is a rigid size variable computed inside f itself. Clearly this makes no sense. I think the rule is that it should be a type error for a rigid size variable to propagate into the type of something that is being let -bound.

Another problem brought about by rigid size variables is the potential for time travel. Specifically, since Futhark allows term-level values to be extracted from types (remember length from above), I am concerned that a type-based unification performed inside a control-flow path ( if branch) that is not executed at run-time, can influence the size of some array generated outside the if . Suppose first that the function concat has the following type:

val concat [n] [m] 'a = [n]a -> [m]a -> []a

Note that if we call concat with two-dimensional arrays, the row sizes must match exactly, because they are identified by the same type parameter ( a ).

Now ponder the following program:

let shape [n] [m] (xss: [n][m]i32) = (n, m) shape [n] [m] (xss: [n][m]i32) = (n, m) let f (b: bool) (xs: []i32) = f (b: bool) (xs: []i32) = let arr = [] -- Inferred as type [][]i32. arr = [] in if b then shape (concat [filter p xs] arr) shape (concat [filter p xs] arr) else shape arr shape arr

What should f false [1,2,3] return? Due to the concat , we are forcing the row size of arr to be the size of filter p xs , which is a rigid size variable. However, if b is false, then filter p xs will never be run, yet we still inspect the size of arr ! I’m not quite sure how to handle this. Either we need to make arr properly polymorphic (currently only functional bindings are let-generalised), or we need to institute rules for certain syntactic constructs (mostly array literals and related cases) that prevent unification with rigid size variables. The fundamental problem is that at run-time, we may need to construct an actual multidimensional array right now, and when that happens, we better have actual sizes for the dimensions available, and not just a promise that they’ll match the size of some other array that may or may not be computed in the future.

As a final problem, consider the type of map :

val map [n] 'a 'b : (a -> b) -> [n]a -> [n]b map [n] 'a 'b : (a -> b) -> [n]a -> [n]b

Should map iota is be well typed? Probably not, since this would only work if all values of is are the same. But how should this be prevented? Maybe type checking of function application should take a closer look at whether any dependent typing is going on in case of higher-order functions, but a clear rule has not yet crystallised.

I don’t think ironing out the kinks in size types is an insurmountable problem. Fundamentally, I think they are quite simple, and I’m perfectly willing to give up some completeness (and resorting to size coercions) in order to maintain simplicity and type inference. However, to fully convince myself and others, we may have to break open the big box of Greek letters, and actually prove some properties about the type system. I’ll get around to that once I figure out exactly what the rules are even supposed to be.

Promisingly, it took fairly little work to make sure most of our benchmarks type-check under the current prototype of size types. I had to fix only a few things, and in most cases I would say they even improved the style of the program. In a few cases, I even fixed real (but dormant) bugs. I think this is because size types more or less just codify what is already good Futhark coding practice.

There is of course also the risk that the type checker is simply buggy and is letting incorrect programs slip by. That’s the frustrating part about working on a type checker: your programs may type check, but that does not mean that the type checker is correct…