I recently fixed a bug in the Futhark type checker (or type rules - this distinction is pretty blurry when the language is practically defined by its implementation). At first I feared a deep conceptual problem that would be difficult to fix, but the eventual solution ended up being fairly clean and simple. I think the entire process is a good case study of what such bugs look like, and so I’m writing up the problem and its solution for those readers who just can’t get enough of hearing about other people’s buggy compilers. I’m also writing it up as a warning against the complexities of substructural type systems - avoid implementing them if at all possible (and as we shall see, the Futhark type system has some quirks that make the situation even worse).
Futhark is a language in the ML family, and its type system is mostly conventional. Parametric polymorphism and the like behaves as one would expect. But there are some novelties. Perhaps the most unusual one is a form of uniqueness types, which we added to support certain important patterns in high-performance array programming. For example, the expression
A with [i] = x produces a the array
A, except that the value at index
i has been replaced by
x. However, it is quite inefficient to copy all of
A just to modify a single value. Instead, the Futhark type rules require that the original array
A is not used afterwards. This enables the compiler to implement the expression as an in-place update of whatever memory
A happens to be stored in, even though semantically, there are no side effects. However, the requirement that
A is “not used afterwards” can get pretty subtle. Here’s a simple example:
let B = A let C = A with [i] = X
B is just another name for
A, we also must not use
B afterwards. By defining various aliasing rules, the type checker can track how different variables may (potentially) refer to the same underlying “value”, and when a value is then consumed in a
with-expression, we mark all of the aliases as similarly consumed and inaccessible. This is necessarily conservative:
let C = if p then A else B let D = C with [i] = x
In the above we must treat
C as potentially aliasing both
B - hence neither of these may be used after the
with-expression. This is slightly complicated to implement in a type checker, but tractable. But then what about the following?
let B = f A let C = A with [i] = X
B is produced by applying some function
f. Does the result alias
A? Does it alias some other random value that might exist somewhere? Who knows! Well, the type checker will have to know. This is precisely the problem that is solved by uniqueness types, by having each function encode which of its arguments it consumes, and which of its return values are guaranteed to be unique (have no aliases). For example, the type of
reverse in Futhark is as follows:
val reverse [n] 't : [n]t -> [n]t
Given an array of
ts (whatever that is), it returns another array of
reverse function does not consume its input, meaning we can
reverse an array and then still use it afterwards, but it also does not return a unique array, meaning that the result of
reverse will be considered to alias its arguments. In contrast, this is the type of
val copy 't : t -> *t
Given any value of some type
copy returns a unique
t (that’s what the asterisk means), implying that the result is guaranteed to be free of aliases. Now consider a function
val update [n] 't : *[n]t -> i64 -> t -> *[n]t
This function takes a unique array as its first argument, and also returns a unique array. This means that after an application
update A i x, the array
A will be considered consumed, and must not be used again - just in a
Once you get used to them, these rules provide a fairly simple way of writing code with an imperative cost model, yet purely functional semantics. Importantly, they are opt-in - if you don’t need this feature, you can mostly ignore these details (except that type inference may sometimes put mysterious asterisks on your return types, when it figures out that a function result will never have aliases).
Futhark’s implementation of uniqueness types is a substructural and non-syntactic type system, which in practical terms means that it can be really tricky to implement correctly. In particular the tracking of transitive aliases is not just an implementation challenge, but it can also sometimes be difficult to explain to the user why something is not allowed:
let B = A let C = B let D = A with [i] = x
After the above,
C must be treated as consumed, as
A, which is consumed in the
But still, just track the names, right? Well, even though this type system feature has been around since before Futhark was called Futhark, we still discover fundamental and slightly embarrassing bugs in our implementation from time to time. The latest bug concerns the case when there are no names to track:
let dup x = (x,x) let main (xs: i32) : (*i32, *i32) = dup (copy xs)
Here we have a function
dup that duplicates its input. It has the following type:
val dup 't : t -> (t,t)
Nothing unexpected - it doesn’t consume its argument, and it returns two non-unique results, meaning that they will be assumed to alias the argument.
main function takes a non-unique array as input and expects to return two unique arrays. It does this by first copying
xs to remove all aliases (fine) and then duplicates the resulting alias-free array, producing two alias-free arrays, as far as the type checker is concerned. But clearly things have gone awry -
dup does not produce two copies, it just produces the same value twice. This should not type check! Let us try rewriting the
main function a bit:
let main (xs: i32) : (*i32, *i32) = let tmp = copy xs in dup tmp
Now the type checker will construct an array
tmp with no aliases (fine), and the call to
dup will then produce two arrays, both of which alias
tmp (and hence each other). When we then claim that these two arrays, which alias each other, constitute two unique arrays, the type checker will complain:
A unique tuple element of return value of "main" is aliased to some other tuple component.
So the problem is that nameless intermediate results inhibit our ability to track aliases correctly, because alias tracking is based on names! If something does not have a name, then it is invisible. That’s bad. In this case, as in many others that involve holes in our type checker, we were actually saved by the Futhark compiler’s design, where the core language is (almost) as tightly typed as the source language, and assigns names to all intermediate results. This defence in depth has proven extremely useful in avoiding code generation bugs (most become compiler crashes instead), but bugs are still bugs, and should be fixed. Anything that passes the source language type checker only to crash in the core language is our fault, not the users’.
If the problem is that intermediate results do not have names, then the solution is intuitively obvious: give them names. Specifically, after every function application, we invent a “fictional” name that does not correspond to any variable in the program, and add it to the aliasing of the result type. In effect, we pretend that every function application is
let-bound, but we don’t actually change the AST. This allows us to track that the two values produced by
dup both alias the same underlying value, and thus reject the program.
Done! Well, almost. This solves the soundness issue, but there’s more to a compiler than just avoiding errors.
Consider this program, which suffers from a very similar problem:
let main n = let (a,b) = let y = iota n in (y,y) let a = a with  = 0 let b = b with  = 0 in (a,b)
Here the problem is that
b both alias
y, so we should not be allowed to consume both of them, but
y has gone out of scope by the time the
with-expressions occur. We address this by using the “fictional” name to track the aliases, but the error message suffers:
Consuming "internal_app_result", but this was previously consumed at 4:7-10.
Compiler errors that just vomit out some internal name or data structure are not exactly rare in research compilers, but it is a rare programmer who likes to be faced with them (except the compiler developers themselves, who will be pleased to be shown exactly what is wrong!).
To improve matters, I added a table that maps these names to a description of the application that gave rise to them, which is then used to synthesise a error message:
Consuming result of applying "iota" (at 2:23-28), but this was previously consumed at 4:7-10.
The phrasing is perhaps clumsy, but at least it is relatively clear, and points to the source of the problem. The idea of using a mapping from compiler-generated names to “reasons” for their existence was first used to make size type errors less incomprehensible, and I think it works pretty well. Futhark’s error messages are still not as good as those produced by some other compilers, but given our limited resources, I think they are pretty decent.