Haskell has exceptions. Therefore, any design for linear types in Haskell will have to deal with exceptions. It might seem impossible to square with this requirement: how can linear types, which require that values be used exactly once, accommodate exceptions, which interrupt my computation?

The mantra to remember, to dispel this feeling, is that “a function f is linear when: if its result is consumed exactly once, then its argument is consumed exactly once”. If the result is consumed multiple times, then the argument is consumed multiple times, if the result is consumed only partially (e.g. because an exception was thrown), then the argument may be consumed only partially (including not at all).

This may, at first, read as a carefully worded misdirection. But I assure you that it isn't. This conditional statement was already a property of linear logic when it was introduced in 1987. And it is indeed one of the key intuitions for how Linear Haskell interacts with exceptions.

## Back to basics with monads

The design of Linear Haskell was deliberately chosen to closely follow linear logic. This is due to an observation called the Curry-Howard correspondence, by which types in some programming languages can be read like propositions in logic. This observation has served programming language design well, and is in fact one of the original design principles of Haskell (by the way, did you know what Curry's first name was?). Haskell corresponds to intuitionistic logic, while Linear Haskell corresponds to linear logic.

If we were to design exceptions for a linearly typed language from first principles, we would start from linear logic and add exceptions on top. The methodology to do so was given to us by Eugenio Moggi: apply a monad. Indeed neither intuitionistic logic nor linear logic have a native notion of exceptions. In fact, when viewed as programming languages using the Curry-Howard correspondence, they only have terminating computations. But Moggi showed how to use monads to model effectful computations.

This has since become routine in Haskell programming, and you probably guessed where this is going: the simplest model of exception, for intuitionistic logic, is the Maybe monad.

However, in linear logic, there are several refinements of the Maybe type. Which one models exceptions? Interestingly enough, not the one called Maybe in Linear Haskell (_⊕1 in linear logic). In fact, if you read my previous blog post, you will know already that it isn't the right kind of monad to apply here. Instead we need to use the _⊕⊤ monad from linear logic, where ⊤ can be defined in Linear Haskell as

newtype Top = Top (forall b. Void #-> b)


The difference between Top and () (aka 1 in the linear logic literature) is that there is a linear function a #-> Top for every type a:

swallow :: a #-> Top
swallow x = Top \$ \v -> case v of {}


But, swallow doesn't use its argument! How can it be a linear function? Remember the mantra: “a function f is linear when if its result is consumed exactly one, then its argument is consumed exactly once”. The trick is that there is no way, in linear logic, to consume a value of type ⊤ exactly once. Indeed neither in Linear Haskell is there a way to consume a value of type Top exactly once (since there is no way to apply the wrapped function exactly once). So, vacuously, swallow will consume its argument exactly once when its result is consumed exactly once.

We can use the type Either Top a to model in Haskell potentially failing computations that return a value of type a. Using swallow, we can write a computation that errors out, ignoring all the linearity requirements which we would otherwise need to honour. Since we can't consume values of type Top exactly once, we can't catch exceptions in a linear computation. We will need an unrestricted computation instead:

catch :: Either Top a -> a -> a
catch (Left _) handler = handler
catch (Right x) _ = x


So the Curry-Howard correspondence, the bridge between logic and programming, compels us to have a catch without linear arguments. Intuitively, this prevents linear variables from escaping outside of the catch.

## Resource management

Now that we know how to model exceptions in linear functions, let us turn to how exceptions interact with applications of linear types. More specifically, how do exceptions interact with resource management?

Resource management is one of the original motivations for Linear Haskell. In a recent blog post, we described, for instance, how we use linear types to manage references across two different garbage collectors to avoid memory leaks in inline-java applications.

The point of linear types for resource management is that types force us to call the release function on our resources to free them, allowing for precise, yet safe, management of the resource. We just can't forget to call release. But it may seem that exceptions throw a wrench in this plan: since we can interrupt the computation at any time, we can entirely bypass the call to release.

However, Linear Haskell doesn't have any notion of resource or resource management. Linear Haskell is only a type system, and doesn't extend the compiler with new concepts. Quite the contrary: the philosophy of Linear Haskell is to empower the programmer to add new abstractions in user space (i.e. without requiring additional compiler support).

So the question isn't, does Linear Haskell correctly manage resources, but instead: is it possible to write a resource management abstraction in Linear Haskell? And to this, the answer is an emphatic yes.

I can make such an unabashed claim because, quite simply, I wrote one. The high-level view is:

• there is the resource monad RIO in which resources are managed,
• each resource type has acquire and release functions,
• resources are linear values,
• the function run :: RIO (Unrestricted a) -> IO a (notice the unrestricted arrow), is responsible for itself releasing all remaining resources if an exception occurs.

This way, resource management is entirely under the programmer's control: resources are released when the appropriate function is called. The programmer can't forget to release a resource, or use it after releasing, since it is prevented by the type system. If an exception occurs, then all the resources are cleaned immediately.

To reiterate, saying that linear types are exception-safe, or on the contrary exception-oblivious, doesn't make sense. Whether system resources are always released in a timely manner even in the face of exceptions is a property of the abstraction you create using linear types to enable programmers to have full but safe control over resources. The resource monad is one such abstraction, just like the Either Top monad is an abstraction, with different properties, for modeling potentially failing computations. The same is true of virtually any other type system feature like higher-ranked types, GADTs, dependent types: they are tools to build abstractions with desirable properties.

It is tempting to say that since computations can be interrupted by exceptions, this system is an affine type system. This is misleading. An affine function is such that if its result is used exactly once then its argument is used at most once.

It's a very different system. For instance, for resource management, there is no guarantee that a resource is released before the entire computation has ended. We could be waiting a long time. It's harder to create an abstraction that provides the same properties as above.

There is indeed some connection between affine types and exceptions. Most notably, catch can be affine in an affine type system.

But wrapping linear logic in the Either Top monad doesn't make it affine. In fact, affinity is not a monadic effect: it's a comonadic coeffect. But this is a story for another time. If you grab me over tea, you can easily get me to talk about it.

## Conclusion

I hope to have convinced you that the interaction of linearity and exceptions in Linear Haskell, as it is currently designed, is not only reasonable: it is natural and necessary.

It doesn't mean that exceptions are free: when writing a new linear abstraction using unsafe functions (such as the FFI), it is your responsibility to ensure that the functions you write are indeed linear—just as when using unsafePerformIO, you need to make sure that the computation really is pure. And when doing so, you need to be mindful of exceptions, which can complicate the implementation. But Haskell has exceptions, and this complication is unavoidable.