[chas]

This is a great paper, but I found it to be very challenging to read when I first ran into it, even though I had some experience programming in Haskell. As a one sentence pitch, this paper is to recursion what if and while are to goto. For a more detailed intro to the concepts see: https://blog.sumtypeofway.com/posts/introduction-to-recursio... If you are comfortable with Haskell, it should be relatively accessible. If you aren’t, but want a bit more flavor than my one sentence, the intro section in that blog post is still worth a read.

[chombier]

I also remember struggling with this paper quite a bit at first. This is how I would explain catamorphisms to past me:

Suppose you have a recursive `tree` data structure, and another one `tree' x` where `tree' x` is just like `tree`, but with recursive occurrences of `tree` replaced with type x. So maybe something like, in Haskell-ish syntax:

    type tree = Leaf int | Node tree tree
    type tree' x = Leaf int | Node x x

Then one can easily write a `fold` over trees with type

    fold: tree -> (tree' x -> x) -> x

where the second parameter (called an algebra) evaluates the result over a tree' assuming recursive subtrees have been replaced with the result of their evaluation (this is the key).

Now the beauty of recursion schemes is that whenever `tree'` is a functor, you can get `tree` (as a fixpoint: `tree = tree' tree`) and `fold` for free (generally called `cata`), which can be seen by rewriting `fold` above using just `map`.

This skips a fair amount of boilerplate code you no longer have to touch when modifying recursive types.

[solomonb]

Minor nitpick, you don't get `fold` as written for free. You would still need to provide a function `toTreeF :: tree -> tree'` then compose that with `cata`, which you do get for free: `fold t alg = cata alg (toTreeF t)`.

[sfvisser]

Isn’t toTreeF just ‘out’ as in ‘newtype Fix f = In { out :: f (Fix f) }’ ?

[solomonb]

Sorry my last comment was sort of wrong because I neglected to mention Fix (I'm going to claim that I was tired :) ).

They say a picture is worth a thousand words, so hopefully this clears up the difference betwene `out` and `toTreeF`:

    -- | Our basic binary tree type
    data Tree a = Leaf a | Node (Tree a) (Tree a)
    
    -- | A pattern functor for our binary tree
    data TreeF a r = LeafF a | NodeF r r
    
    -- | The fixed point of 'f'.
    newtype Fix f = Fix { unfix :: f (Fix f) }
    
    -- | Given an F-Algebra on 'f' we have a catamorphism on 'Fix f'
    cata :: Functor f => Algebra f a -> Fix f -> a
    cata alg fix = alg $ fmap (cata alg) $ unfix fix
    
    -- | We can sum the elements of a 'Fix (TreeF Int)' using 'cata'
    sumTreeF :: Fix (TreeF Int) -> Int
    sumTreeF = cata $ \case
      LeafF i x -> x + i
      NodeF l r x -> x + l + r 
    
    -- | To operate our binary tree we need to map from 'Tree a' to 'TreeF
    -- a'.
    toTreeF :: Tree a -> Fix (TreeF a)
    toTreeF = \case
      Leaf a -> LeafF a
      Node l r -> NodeF (toTreeF l) (toTreeF r)
    
    sumTree :: Tree Int -> Int
    sumTree = sumTreeF . toTreeF

I didn't compile this code but it should be correct.

    out :: Fix (TreeF a) -> TreeF a (Fix (TreeF a))
    toTreeF -> Tree a -> Fix (TreeF a)

[chombier]

If you're going to introduce a generic functor fixpoint then you'll probably want to define

     type Tree a = Fix (TreeF a)

so that `toTreeF` is just `unfix` (and is effectively free, as I mentioned above)

[solomonb]

Yeah you could do that but now you have thrown away the recursive `Tree` type and are always working in `TreeF`. This is totally fine but is a different design decision from where we started.

[cloogshicer]

Thanks for explaining!

[jimbokun]

If I understand the idea of the linked article correctly, I sometimes do a similar thing in Java by defining an Iterator that implements some sort of traversal over a recursive data structure, then can apply any operations to the returned items using a for loop, applying an anonymous function to all the items, etc.

From some of the sibling comments, seems like there is a strong connection between these recursion schemes and iterator strategies?

[platz]

so its structured recursion i.e. it reifies recursion into variables that than be passed around and modified instead of being implicit/embedded in the control flow?

if so how is that different from Observable streams or event emitters.

[grumpyprole]

The .NET RX Observable streams where designed by Erik Meijer the author of this paper. The GOF book came up with the "Observer" pattern, but failed to spot and exploit the duality with iterators. RX and LINQ both take ideas from functional programming.

[platz]

> The GOF book came up with the "Observer" pattern, but failed to spot and exploit the duality with iterators.

That's an interesting insight - thanks

[chas]

While the first author of this paper went on to be heavily involved in Microsoft's Reactive Extensions (among many other things), I think it's better to think of it as making common recursive patterns explicit and introducing abstraction for those patterns. For example, a common use of goto is to run a certain block of code until a condition becomes true. This is abstracted into a pattern called "while" which takes a condition and a block of code to run. A common use of recursion is the reduce (also called fold) operation which, in the case of a list, computes a single value from a list by combining all of the values of the list with a binary function. As an example, this code sums all of the values in an integer list:

  def sum(xs):
    if xs == []:
      return 0
    else:
      x = xs.pop(0)
       return x + sum(xs)

This sort of reduction operation is very broadly useful, so it has been abstracted in frameworks like MapReduce as well as library functions like functools.reduce (https://docs.python.org/3/library/functools.html#functools.r...). Recursion schemes build on explicit reduce functions by being strongly typed (which, in addition to reducing bugs, enables a something like a super-powered visitor pattern), very orthogonal (which reduce redundancy and code duplication as a user of the abstraction), and very general (which let you solve a lot of problems with the same small set of programming tools without needing to remember special cases). In particular, the way that they look at data structures lets you interleave the recursion across nested data structures in a way that would be a huge pain in the butt with e.g. Python's __iter__ interface. There are some other nifty things that the approach brings to the table as well, but I think those are the major wins from the perspective of someone not already interested in strongly-typed functional programming.

While I covered the case of things that are sort of like reduce (called catamorphisms in the language of recursion schemes), this paper also has analogous abstractions for things that are sort of like itertools.accumulate (https://docs.python.org/3/library/itertools.html#itertools.a..., called anamorphisms in recursion schemes), as well as combinations thereof (called hylomorphisms). They all use a relatively small number of building blocks and combine in a quite beautiful and useful way, but it's hard to describe precisely without leaning quite heavily on the language of strongly-typed functional programming.

[solomonb]

I can't speak to Observable streams but one of the nicest things about recursion schemes is that they are guaranteed to terminate.