Review: Bananas, Lenses, Envelopes and Barbed Wire

Today’s classic functional programming paper we will review is Meijer et al.’s Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire. The exciting gist of the paper is that all explicit recursion can be factored out into a few core combinators. As such, the reasoning is that we should instead learn these combinators (or “recursion schemes” as they’re called), rather than doing our own ad-hoc recursion whenever we need it.

Despite being a marvelous paper, it falls into the all-too-common flaw of functional programming papers, which is to have an absolutely horrible title. “Bananas”, “lenses”, “envelopes” and “barbed wire” correspond to obscure pieces of syntax invented to express these ideas. In our treatment of the literature, we will instead use standard Haskell syntax, and refer to the paper as Functional Programming with Recursion Schemes.

Specialized Examples of Recursion Schemes

Catamorphisms over Lists

Catamorphisms refer to a fold over a datastructure. A mnemonic to remember this is that a catamorphism tears down structures, and that if that structure were our civilization it’d be a catastrophe.

By way of example, Meijer et al. present the following specialization of a catamorphism over lists:

Let default :: b and step :: a -> b -> b, then a list-catamorphism h :: [a] -> b if a function of the following form:

h [] = default
h (a : as) = step a (h as)

This definition should look pretty familiar; if you specialize the function foldr to lists, you’ll see it has the type:

foldr :: (a -> b -> b) -> b -> [a] -> b

We can view foldr as taking our values step :: a -> b -> b and default :: b, and then giving back a function that takes an [a] and computes some b. For example, we can write a few of the common prelude functions over lists as catamorphisms of this form.

length :: [a] -> Int
length = foldr (\_ n -> n + 1) 0
filter :: forall a. (a -> Bool) -> [a] -> [a]
filter p = foldr step []
  where
    step :: a -> [a] -> [a]
    step a as = if p a
                   then a : as
                   else as

When written this way – Meijer et al. are quick to point out – the so-called “fusion” law is easily seen:

f . foldr step default = foldr step' default'
  where
    step' a b = step a (f b)
    default'  = f default

which intuitively says that you can “fuse” a catamorphism with a subsequent composition into a single catamorphism.

Anamorphisms over Lists

If a catamorphism refers to a “fold”, an anamorphism corresponds to an unfold of a data structure. A good mnemonic for this is that an anamorphism builds things up, just like anabolic steroids can be an easy way to build up muscle mass.

Meijer et al. present this concept over lists with the following (again, very specialized) definition:

unfoldr :: (b -> Maybe (a, b)) -> b -> [a]

Given a function produce :: b -> Maybe (a, b), a list-anamorphism h :: b -> [a] is defined as:

h seed = case produce seed of
           Just (a, b) = a : h b
           Nothing     = []

As expected, this corresponds to the unfoldr :: (b -> Maybe (a, b)) -> b -> [a] function from Data.List.

By way of example, they provide the following:

zip :: ([a], [b]) -> [(a, b)]
zip = unfoldr produce
  where
    produce (as, bs) =
      if null as || null bs
         then Nothing
         else Just ((head as, head bs), (tail as, tail bs))

and

iterate :: (a -> a) -> a -> [a]
iterate f = unfoldr (\a -> Just (a, f a))

An interesting case is that of map :: (a -> b) -> [a] -> [b]. We note that both the input and output of this function are lists, and thus might suspect the function can be written as either a catamorphism or an anamorphism. And indeed, it can be:

cataMap :: (a -> b) -> [a] -> [b]
cataMap f = foldr (\a bs -> f a : bs) []

anaMap :: (a -> b) -> [a] -> [b]
anaMap f = unfoldr produce
  where
    produce [] = Nothing
    produce (a : as) = Just (f a, as)

Neat!

Hylomorphisms over Lists

A hylomorphism over lists is a recursive function of type a -> b whose call-tree is isomorphic to a list. A hylomorphism turns out to be nothing more than a catamorphism following an anamorphism; the anamorphism builds up the call-tree, and the catamorphism evaluates it.

An easy example of hylomorphisms is the factorial function, which can be naively (ie. without recursion schemes) implemented as follows:

fact :: Int -> Int
fact 0 = 1
fact n = n * fact (n - 1)

When presented like this, it’s clear that fact will be called a linear number of times in a tail-recursive fashion. That sounds a lot like a list to me, and indeed we can implement fact as a hylomorphism:

fact :: Int -> Int
fact = foldr (*) 1 . unfoldr (\n -> if n == 0
                                       then Nothing
                                       else Just (n, n - 1))

The hylomorphic representation of fact works by unfolding its argument n into a list [n, n-1 .. 1], and then folding that list by multiplying every element in it.

However, as Meijer et al. point out, this implementation of fact is a little unsatisfactory. Recall that the natural numbers are themselves an inductive type (data Nat = Zero | Succ Nat), however, according to the paper, there is no easy catamorphism (nor anamorphism) that implements fact.

Paramorphisms

Enter paramorphisms: intuitively catamorphisms that have access to the current state of the structure-being-torn-down. Meijer et al.:

[Let init :: b and merge :: Nat -> b -> b. ] For type Nat, a paramorphism is a function h :: Nat -> b of the form:

h Zero = init
h (Succ n) = merge n (h n)

As far as I can tell, there is no function in the Haskell standard library that corresponds to this function-as-specialized, so we will write it ourselves:

paraNat :: (Nat -> b -> b) -> b -> Nat -> b
paraNat _ init Zero = init
paraNat merge init (Succ n) = merge n (paraNat merge init n)

We can thus write fact :: Nat -> Nat as a paramorphism:

fact :: Nat -> Nat
fact = paraNat (\n acc -> (1 + n) * acc) 1

Similarly, we can define paramorphisms over lists via the function:

paraList :: (a -> [a] -> b -> b) -> b -> [a] -> b
paraList _ init [] = init
paraList merge init (a : as) = merge a as (paraList merge init as)

with which we can write the function tails :: [a] -> [[a]]:

tails :: forall a. [a] -> [[a]]
tails = paraList merge []
  where
    merge :: a -> [a] -> [[a]] -> [[a]]
    merge a as ass = (a : as) : ass

General Recursion Schemes

Intuition

As you’ve probably guessed, the reason we’ve been talking so much about these recursion schemes is that they generalize to all recursive data types. The trick, of course, is all in the representation.

Recall the standard definition of list:

data List a = Nil
            | Cons a (List a)

However, there’s no reason we need the explicit recursion in the Cons data structure. Consider instead, an alternative, “fixable” representation:

data List' a x = Nil'
               | Cons' a x

If we were somehow able to convince the typesystem to unify x ~ List' a x, we’d get the type List' a (List' a (List' a ...)), which is obviously isomorphic to List a. We’ll look at how to unify this in a second, but a more pressing question is “why would we want to express a list this way?”.

It’s a good question, and the answer is we’d want to do this because List' a x is obviously a functor in x. Furthermore, in general, any datastructure we perform this transformation on will be a functor in its previously-recursive x parameter.

We’re left only more curious, however. What good is it to us if List' a x is a functor in x? It means that we can replace x with some other type b which is not isomorphic to List a. If you squint and play a little loose with the type isomorphisms, this specializes fmap :: (x -> b) -> List' a x -> List' a b to (List a -> b) -> List' a x -> List' a b.

Notice the List a -> b part of this function – that’s a fold of a List a into a b! Unfortunately we’re still left with a List' a b, but this turns out to be a problem only in our handwaving of x ~ List' a x, and the actual technique will in fact give us just a b at the end of the day.

Algebras

An f-algebra is a function of type forall z. f z -> z, which intuitively removes the structure of an f. If you think about it, this is spiritually what a fold does; it removes some structure as it reduces to some value.

As it happens, the type of an f-algebra is identical to the parameters required by a catamorphism. Let’s look at List' a-algebras and see how the correspond with our previous examples of catamorphisms.

length :: [a] -> Int
length = foldr (\_ n -> n + 1) 0

lengthAlgebra :: List' a Int -> Int
lengthAlgebra Nil'        = 0
lengthAlgebra (Cons' _ n) = n + 1
filter :: forall a. (a -> Bool) -> List a -> List a
filter p = foldr step Nil
  where
    step a as =
      if p a
         then Cons a as
         else as

filterAlgebra :: (a -> Bool) -> List' a (List a) -> (List a)
filterAlgebra _  Nil'        = Nil
filterAlgebra p (Cons' a as) =
  if p a
    then Cons a as
    else as

Coalgebras

f-algebras correspond succinctly to the parameters of catamorphisms over fs. Since catamorphisms are dual to anamorphisms, we should expect that by turning around an algebra we might get a representation of the anamorphism parameters.

And we’d be right. Such a thing is called an f-coalgebra of type forall z. z -> f z, and corresponds exactly to these parameters. Let’s look at our previous examples of anamorphisms through this lens:

zip :: (List a, List b) -> List (a, b)
zip = unfoldr produce
  where
    produce (as, bs) =
      if null as || null bs
         then Nothing
         else Just ((head as, head bs), (tail as, tail bs))

zipCoalgebra :: ([a], [b]) -> List' (a, b) ([a], [b])
zipCoalgebra (as, bs) =
  if null as || null bs
     then Nil'
     else Cons (head as, head bs) (tail as, tail bs)
iterate :: (a -> a) -> a -> [a]
iterate f = unfoldr (\a -> Just (a, f a))

iterateAlgebra :: (a -> a) -> a -> List' a a
iterateAlgebra f a = Cons a (f a)

You might have noticed that these coalgebras don’t line up as nicely as the algebras did, due namely to the produce functions returning a type of Maybe (a, b), while the coalgebras return a List' a b. Of course, these types are isomorphic (Nothing <=> Nil', Just (a, b) <=> Cons a b), it’s just that the authors of unfoldr didn’t have our List' functor to play with.

From Algebras to Catamorphisms

As we have seen, f-algebras correspond exactly to the parameters of a catamorphism over an f. But how can we actually implement the catamorphism? We’re almost there, but first we need some machinery.

type family Fixable t :: * -> *

The type family Fixable takes a type to its fixable functor representation. For example, we can use it to connect our List a type to List' a:

type instance Fixable (List a) = List' a

Now, assuming we have a function toFixable :: t -> Fixable t t, which for lists looks like this:

toFixable :: List a -> Fixable (List a) (List a)
-- equivalently: toFixable :: List a -> List' a (List a)
toFixable Nil         = Nil'
toFixable (Cons a as) = Cons' a as

We can now write our catamorphism!

cata :: (Fixable t z -> z) -> (t -> z)
cata algebra = algebra
             . fmap (cata algebra)
             . toFixable

Very cool. What we’ve built here is general machinery for tearing down any inductive data structure t. All we need to do it is its Fixable t representation, and a function project :: t -> Fixable t t. These definitions turn out to be completely mechanical, and thankfully, can be automatically derived for you via the recursion-schemes package.

Coalgebras and Catamorphisms

We can turn all of our machinery around in order to implement anamorphisms. We’ll present this material quickly without much commentary, since there are no new insights here.

Given fromFixable :: Fixable t t -> t, we can implement ana:

ana :: (z -> Fixable t z) -> z -> t
ana coalgebra = fromFixable
              . fmap (ana coalgebra)
              . coalgebra

General Paramorphisms

Because there is nothing interesting about hylomorphisms when viewed via our Fixable machinery, we skip directly to paramorphisms.

Recall that a paramorphism is a fold that has access to both the accumulated value being constructed as well as the remainder of the structure at any given point in time. We can represent such a thing “algebraically:” Fixable t (t, z) -> z. With a minor tweak to cata, we can get para:

para :: (Fixable t (t, z) -> z) -> t -> z
para alg = teardown
  where
    teardown = alg
             . fmap (\t -> (t, teardown t))
             . toFixable

Miscellaneous Findings

All Injective Functions are Catamorphisms

Meijer et al. make the somewhat-unsubstantiated claim that all injective functions are catamorphisms. We will reproduce their proof here, and then work through it to convince ourselves of its correctness.

Let f :: A -> B be a strict function with left-inverse g. Then for any φ :: F A -> A, we have:

g . f = id

f . cata φ = cata (f . φ . fmap g)

Taking `φ = fromFixable$ we immediately get that any strict injective function can be written as a catamorphism:

f = cata (f . fromFixable . fmap g)

Sounds, good? I guess? Meijer et al. must think I’m very smart, because it took me about a week of bashing my head against this proof before I got it. There were two stumbling blocks for me which we’ll tackle together.

To jog our memories, we’ll look again at the definition of cata:

cata φ = φ . fmap (cata φ) . toFixable

There are two claims we need to tackle here, the first of which is that given φ = fromFixable:

f . cata φ = cata (f . φ . fmap g)

We can show this by mathematical induction. We’ll first prove the base case, by analysis of the [] case over list-algebras.

  cata (f . φ . fmap g) []
  -- definition of cata
= f . φ . fmap g . fmap (cata (f . φ . fmap g)) $ toFixable []
  -- definition of toFixable
= f . φ . fmap g . fmap (cata (f . φ . fmap g)) $ NilF
  -- fmap fusion
= f . φ . fmap (g . cata (f . φ . fmap g)) $ NilF
  -- NilF is a constant functor
= f . φ $ NilF
  -- substitute φ = fromFixable
= f $ fromFixable NilF
  -- definition of fromFixable
= f []

Great! That’s the base case tackled. It’s easy to see why this generalizes away from lists to any data structure; the only way to terminate the recursion of a cata is for one of the type’s data constructors to not be recursive (such as Nil). If the data constructor isn’t recursive, its Fixable representation must be a constant functor, and thus the final fmap will always fizzle itself out.

Let’s tackle the inductive case now. We’ll look at cases of cata that don’t fizzle out immediately:

  cata (f . φ . fmap g)
  -- definition of cata
= f . φ . fmap g . fmap (cata (f . φ . fmap g)) . toFixable
  -- fmap fusion
= f . φ . fmap (g . cata (f . φ . fmap g)) . toFixable
  -- definition of cata
= f . φ . fmap (g . f . φ . fmap g . fmap (cata (f . φ . fmap g)) . toFixable) . toFixable
  -- fmap fusion
= f . φ . fmap (g . f . φ . fmap (g . cata (f . φ . fmap g)) . toFixable) . toFixable
  -- g . f = id
= f . φ . fmap (id . φ . fmap (g . cata (f . φ . fmap g)) . toFixable) . toFixable
  -- definition of id
= f . φ . fmap (φ . fmap (g . cata (f . φ . fmap g)) . toFixable) . toFixable

As you can see here, subsequent expansions of cata line their gs and fs up in such a way that they cancel out. Also, we know from our experience looking at the base case that the final g will always sizzle out, and so we don’t need to worry about it only being a left-inverse.

The other stumbling block for me was that cata fromFixable = id, but this turns out to be trivial:

  cata fromFixable
= fromFixable . fmap (cata fromFixable) . toFixable

Eventually this will all bottom out when it hits the constant functor, which will give us a giant chain of fromFixable . toFixables, which is obviously id.

To circle back to our original claim that all injective functions are catamorphisms, we’re now ready to tackle it for real.

f . cata φ           = cata (f . φ . fmap g)
f . cata fromFixable = cata (f . fromFixable . fmap g)
f . id               = cata (f . fromFixable . fmap g)
f                    = cata (f . fromFixable . fmap g)

\(\blacksquare\)

All Surjective Functions are Anamorphisms

Anamorphisms are dual to catamorphisms, and surjective functions are dual (in \(\mathbb{Set}\)) to injective functions. Therefore, we can get this proof via duality from the proof that injective functions are catamorphisms. \(\blacksquare\)

Closing Remarks

Functional Programming with Recursion Schemes has some other (in my opinion) minor contributions about this stuff, such as how catamorphisms preserve strictness, but I feel like we’ve tackled the most interesting pieces of it. It is my hope that this review will serve as a useful complement in understanding the original paper.