On Free DSLs and Cofree interpreters
This post has been triggered by a tweet from Eric Torreborre on a talk by David Laing presenting the interaction of Free DSLs and Cofree interpreters at the Brisbane Functional Programming Group. I am currently engaged in the development of a Haskellbased system for Capital Match which is basically an API for managing peertopeer lending, and I am trying to formalise the API of the system as the result of a composition of several domainspecific languages.
The ultimate goal is to be able to use these DSLs to define complex actions that could be interpreted in various ways: a commandline client sending RESTful queries to a server, a Webdriverbased test executor or a simple test recorder and comparator, or even by a core engine interpreting complex actions in terms of simpler sequencing of service calls.
The rest of the post is a simple literate Haskell style explanation of what I came up with today exploring the specific topic of the composition of DSLs and interpreters: Given we can compose DSLs using Free monads and Coproduct, how can we Pair a composite DSL to the composition of several interpreters? The answer, as often, lies in the category theoretic principle for duality: Reverse the arrows! One composes interpreters into a Product type which is then lifted to a Cofree comonad paired to a Free Coproduct monad.
This post has no original idea and is just rephrasing and reshaping of work done by more brilliant people than I am:
 Dan Piponi’s Cofree meets free blog post,
 This thread on Stack overflow about free monads,
 Runar Bjarnason talk on Reasonably Priced Monads,
 An Haskell implementation of the above by Aaron Levin,
 Comonads are objects by Gabriel Gonzalez,
 Data types à la carte by Wouter Swiestra,
 Edward Kmett’s All about comonads slide deck,
 And of course David Laing’s github repository.
I would not dare to say I really understand all of this, but at least I got some code to compile and I have some ideas on how to turn this into a useful “pattern” in our codebase.
Free Coproduct DSLs
So let’s start with some usual declaration and imports…
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This relies on the free package which defines standard free Constructions for Applicative
and Monad
, and cofree for Comonads
.
We define our basic businessdomain specific functors, one for logging some messages and another for persisting some string value. The actual functors defined are not important, what interests us here is the fact we define those “actions” independently but we want in the end to be able to “Assemble” them yielding more complex actions which can at the same time log messages and persist things.
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Our composite DSL should be able to interpret actions which are either logging actions, or persist actions, so we need a way to express this alternative at the typelevel, introducing the notion of Coproduct or Sum. This work has already been packaged by Ed Kmett in the comonadstransformers package but let’s rewrite it here for completeness’ sake.
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A Coproduct
of two functors is then simply the typelevel equivalent of the familiar Either
type, for which we provide smart constructors to inject values from left or right and a suitable Functor
instance.
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We want to be able to implicitly “lift” values from a component into its composite without resorting to explicit packing of the various parts of the alternative formed by a Coproduct
type, something which would be extremely cumbersome to express, hence the introduction of a natural transformation Inject
expressed in Haskell as a typeclass.
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To be useful we provide several interesting instances of this typeclass that defines how to inject functors into a Coproduct
. Note that this requires the OverlappingInstances
extension otherwise the compiler^{1} will refuse to compile our programs. I think this stuff could be expressed as type families but did not manage to get it right, so I gave up and resorted to original formulation by Wouter Swiestra.
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Finally, we provide “smart constructors” that generates Free
monadic expressions out of the individual instructions of our two tiny DSLs. We use a inFree
function combining lifting into Free
monad and possible transformation between functors so that each expressed action is a Free
instance whose functor is polymorphic. This is important as this is what will allow us to combine arbitrarily our DSL fragments into a bigger DSL.
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Equipped with all this machinery we are ready to write our first simple program in a combined DSL:
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Cofree Product Interpreters
We are now done with the DSL part, let’s turn to the interpreter part. First we need some atomic interpreters which should be able to interpret commands from each of our DSL. We will prefix these functors with Co
to demote the relationship they have with the DSL functors. Something which is not obvious here (because our DSL functors only have a single constructor) is that these interpreters should have a dual structure to the DSL functors: Given a DSL expressed as a sum of constructors, we need an interpreter with a product of intepretation functions. The DSL presented in David’s post are more expressive…
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Of course we need concrete interpretation functions, here some simple actions that print stuff to stdout, running in IO
.
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To be able to compose these interpreters we need a Product
type whose definition is straightforward: This is simply the typelevel equivalent of the (,)
tupling operator.
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Then we can define our complex interpreter and what interpretation means in the context of this composite. coiter
is a function from the Cofree
module that “lifts” computation in a Functor into a Cofree
monad, starting from a seed value.
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Tying Free to Cofree
This is where the “magic” occurs! We need a way to tie our DSLs to our interpreters so that we can apply the latter to the former in a consistent way, even when they are composed. Enters the Pairing
class which express this relationship using a function tying together each functor (DSL and interpreter) to produce a result.
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For the Identity
functors, pair
ing is simply twoarguments function application.
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We can also define a pair relating function types and tuple types, both ways:
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And finally we can pair Cofree
and Free
as well as Product
and Coproduct
, thus providing all the necessary tools for tying the knots. Note that in this case no intepretation takes place before pairing hit a Pure
value, which actually means that interpretation first need to build all the “spine” for program to be interpreted then unwind it and applying interpretation step to each instruction. This precludes evaluating infinite “scripts”.^{2}
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We finally tie the appropriate “leaf” functors together in a straightforward way.
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We are now ready to define and interpret programs mixing logging and persistence:
let prog = store “bar” >> logI “foo” >> store “quux” >> logI “baz” :: Free Effect () λ> pair const interpretEffect ((return <$> prog) :: Free Effect (IO ()) ) “storing bar” “foo” “storing quux” “baz” λ>
Conclusion
As is often the case when dealing with “complex” or rather unfamiliar category theoretic constructions, I am fascinated by the elegance of the solution but I can’t help asking “What’s the point?” There is always a simpler solution which does not require all this machinery and solves the problem at hand. But in this case I am really excited about the possibilities it opens in terms of engineering and architecting our system, because it gives us a clear and rather easy way to:
 Define in isolation fragments of DSL matching our APIs and business logic,
 Define one or more interpreter for each of these fragments,
 Combine them in arbitrary (but consistent for pairing) ways.
This code is in gist.