[Haskell-cafe] The relationship between F-algebras and the Free Monad

William Yager will.yager at gmail.com
Sun Aug 14 03:10:15 UTC 2016

Hello all,

I recently read the article "Understanding F-algebras"[1], which goes over
the usage of F-algebras for e.g. evaluating ASTs. This article goes over how

1. Data.Fix creates the initial F-algebra over a functor f.

2. We can express our AST as a non-recursive functor instead of a recursive

3. We can express our evaluator as an Algebra (of the type `f a -> a`,
where f is the type of our AST and `a` is the Algebra's carrier type, i.e.
our result type)

4. We can express terms in our language as members of the type `Fix AST`
(i.e. the initial F-algebra over our AST).

5. By using a catamorphism over our evaluator algebra, we can easily
construct an efficient evaluator.

As a brief example of these principles, see the following program (using

data AST a = Lit Int | Add a a deriving Functor
-- data Fix f = Fix (f (Fix f)) -- defined in Data.Fix
type AST' = Fix AST
program :: AST'
program = Fix $ Add (Fix $ Lit 5) (Fix $ Lit 4)
alg :: AST Int -> Int
alg (Lit i) = i
alg (Add l r) = l + r
eval :: AST' -> Int
eval = Data.Fix.cata alg
main = print $ eval program

This is super cool!

One thing that stood out to me is that there are a number of similarities
here with the usage of the Free Monad. In particular, see "Why free monads
matter" [2]. This article is doing basically the exact same thing. You
express your AST as a non-recursive functor parametrized over the type of
subterms. However, instead of making your terms of type `Fix AST`, they are
of type `Free AST a`.

Now, the recursive constructor for Free is almost identical to the
constructor for Fix, except of course the additional type parameter `a`.
However, both types are of the form `data T = R (f T)`. Free simply has the
additional `Pure` constructor as well.

So obviously, there is some sort of relationship here. However, I don't
have a good enough grasp on these concepts for the relationship to make
itself fully apparent. Some things I've noticed:

Operations on `Fix AST` terminate because the AST type has leaf
constructors without recursion. So a catamorphism over an evaluation
algebra can plausibly terminate because eventually it will hit a leaf node
and the evaluation will stop. On the other hand, operations on `Free AST a`
can also terminate because you might run into a `Pure` constructor. The
difference being that `Pure` is a part of the wrapper type (Free) and not
the wrapped type (AST).

A practical question I have is: Can we use the Free Monad to write AST
terms using `do` notation (useful for DSLs) and then use the elegant
catamorphism evaluator trick (or something similar) to evaluate our ASTs?
Or does evaluating the Free Monad require something more general due to the
Pure constructor? It looks like `Control.Monad.Free.iter` might be
equivalent to `Data.Fix.cata`, but I'm not sure. It looks like a rough
translation works in this case:

data AST a = Add a a deriving Functor -- No more "Const"
type AST' a = Free AST a
program :: AST' Int
program = Free $ Add (Pure 5) (Pure 4)
alg :: AST Int -> Int
alg (Add l r) = l + r
eval :: AST' Int -> Int
eval = Control.Monad.Free.iter alg
main = print $ eval program

So, intuitively, it seems like Free lets us do basically the same thing as
Fix but with the evaluation result present as the last argument of the
expression type.

Sorry for being a bit rambly, and I may also be completely off base here,
so correct me if I'm not making any sense.


[1] https://www.schoolofhaskell.com/user/bartosz/understanding-algebras

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