Sunday, April 22, 2012

Monad transformers are a very effective abstraction mechanism. But, they can make code difficult to reason about, and they sometimes suffer from a form of what C++ developers call the "n times m" problem. That is, if you separate implementation in the form of transformer stacks, from interfaces in the form of typeclasses, you potentially need an instance for each typeclass/transformer combination. The GHC developers have generally avoided transformers1. While, it has been suggested that best practice (at least most of the time) is to define custom monads, either by defining your own machinery, or by wrapping a transformer stack in a newtype.

One of my fellow students suggested that "the right way" to do it was to list the operations you wanted as functor, and then to simply use the free monad over this set of operations. This approach is quite elegantly described in Swierstra's Data types à la carte

Given this discusion, I spent a fair chunk of my weekend thinking about just how far one could go with this approach. It is known that most of the common monad transformers (really everything except Cont) can be encode as free monads. We need something a bit more interesting.

I am bit obsessed with fine grain control of side effects. Perhaps, the most widely used system for fine grained effect control is Java's checked exceptions. Checked exceptions in Java get a bad rap, and with reason. In a language without type inference they are clunky to work with. And, the hierarchical model makes it too appealing to But, Haskell does not have these problems. What's more, Haskell is flexible enough that checked exceptions can be implemented as a library. You can download a checked exception library on Hackage today. In this post we will implement a simple checked exceptions library using an interesting variant of the "À la Carte" approach.

Machinery

First thing is to define some of the machinery we will use for building the exceptions library. We need some extensions we also need one of IncoherentInstances or OverlappingInstances. It does not matter which we use for this post. But, for somethings we will want to do with this machinery we need IncoherentInstances

Right now we are only interested in the supporting machinery We need free monads. There is a library for that.

Data types a la carte heavily uses functor coproducts, as will we.

We also need a way to express that a given functor is subsumed by a coproduct. This is the one spot where we have overlapping instances.

Swierstra's model for how we use these is to define algebras over functors, and build evaluators to "run" free monads by folding over those algebras. We will make special use of one such algebra. The void functor is uninhabited. It is, though, an honest functor. In particular, it is the initial object in the category of endofunctors, and the initial object under functor coproducts, making it surprisingly useful. Often we would need to define a typeclass to encode an interesting algebra, but here we can express it trivially foldFree is just like fold over lists, but over free monads.

So far we haven't deviated much from the norm. Now though, we build some unusual machinery. One of the reasons to prefer free monads over monad transformers is to avoid the linear nature of a stack of monad transformers. Sometimes though, a stack is just what we need. We can recover the stack by assuming that the functor inside a free monad is not an arbitrary tree of coproducts, but rather is one long list (at least it has a list like head) The reason for this is that in addition to the algebraic style, sometimes we want to run one functor at a time

That is all the machinery for "A la Carte" programming. Now, we just have to use it.

Checked exceptions

We don't need much this time as far as extensions the API is small and self documenting Now it is easy. We need a functor to represent exceptions. Since, Maybe is the free monad over the functor that has only one constructor taking no parameters, Either is the free monad over the functor that has only one constructor taking a single arbitrary object as a parameter. raising exception is now trivial catching exceptions is a bit more involved, but still rather simple. The idea is to use the removeStep function to "remove" the appropriate Throws functor from the code we are catching we have to assume that this is the front of the functor co-list.

So now you are probably wondering how this works in practice. Lets get a new file with no extensions all of these work without a hitch. Type inference correctly figures them all out (we can't write the type signatures without turning some extensions on, however). The more interesting test is Which handler gets called? Turns out it is always the first (as we would hope). Interestingly enough, it compiles and uses the first even if we use OverlappingInstances instead of IncoherentInstances.

Prelude Main> testExceptions
["test1","test2, expect e1 Ex1","test3, expect e2 Ex2","test4, expect e1 Ex1","test5, first handler Ex1"]


Next up: memory regions.

1. Norman Ramsey Stack Overflow