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Grokking recursion-scheme: Part 1

Posted on May 19, 2014
Tags: haskell

This post is a little different than the rest of my blog, I’m not nearly as competent with recursion-schemes as I want to be and I don’t understand them fully (yet). This isn’t entirely complete, but I hope it will provide a useful intuition for how to work with some of the lower ends of recursion-schemes and some idea of how to get into the higher end. I’ll be reading this again in two weeks once I’ve forgotten all of this (again). You’ve been warned…

Why Bother?

First, let’s talk about why anyone would care about using a library like recursion-schemes.

Remember back in the good old days when all a programmer was goto and guts? And everyone hated it? We’re at a not dissimilar place in Haskell. Well, it’s not nearly so bad nowadays, however, our principle form of control flow is recursion and really we mostly use recursion in a raw, unprincipled way.

However, we’re starting to move away from it. Do these look familiar?

    foldr :: (a -> b -> b) -> b -> [a] -> b
    foldr f nil (x : xs) = x `f` foldr f nil xs
    foldr f nil []       = nil

foldr is all about abstracting away raw recursion! foldr is great in this way since it covers a surprisingly large cover of cases

    map :: (a -> b) -> [a] -> [b]
    map f = foldr ((:) . f) []

    filter :: (a -> Bool) -> [a] -> [a]
    filter p = foldr (\x rest -> if p x then x : rest else rest) []

Turns out you can implement quite a lot of Data.List with foldr and poor judgment.

However, this isn’t good enough. For example, I do a lot of work with compilers and therefore spend a lot of time doing transformations on trees. I want something like foldr to deal with this.

recursion-schemes is one such option. It’s a way of generalizing these uniform transformations on structures and it’s expanded to cover a lot transformations.

On to recursion-schemes

Now that we know that recursion-schemes is solving a useful problem, let’s get into actually using it. First, we can install it off of hackage

cabal install recursion-schemes

And import everything with

    {-# LANGUAGE TypeFamilies, DeriveFunctor #-}
    import Data.Functor.Foldable

Let’s get started by seeing how recursion-schemes covers foldr

First, we define our own custom list

    data MyList a = MyCons a (MyList a) | MyNil

Next, we define another type of list, with the recursion factored out

    data BList a b = BCons a b | BNil
         deriving Functor

Here b is the recursive bit of BList factored out into an explicit parameter. So

    MyList a ~ BList a (BList a (BList a ....))

The fancy term for this would be to say that List a is the “fixed point” for BList a.

Now we can actually use recursion-schemes

    type instance Base (List a) = BList a

    instance Foldable (List a) where
      project (MyCons a b) = BCons a b
      project MyNil        = BNil

And we’re done. So to understand what’s going on we need to talk about another data type and a little math.

    newtype Fix f a = Fix {unFix :: f (Fix f a)}

Remember before how I mentioned how MyList is the fixed point of BList? Well Fix let’s us exploit this fact. In particular

    out :: Fix (BList a) -> MyList a
    out (Fix (BCons a rest)) = MyCons a (out rest)

    into :: MyList a -> Fix (BList a)
    into (MyCons a rest) = Fix (BCons a $ into rest)

So we could write either BList or MyList for all our data types, but the BList version is really a pain to write since everything is wrapped in Fix. Unfortunately though, it’s much easier to write generic code for stuff of the form Fix (f a).

To solve this recursion-schemes has the type class Base where we map the recursive data type to its non-recursive friend. Then, in project we define how to.. well.. project the recursive into a partially unfolded equivalent.

With just those two steps, we get a large chunk of recursion-schemes operations for our data type!

Just What Did We Get?

Now this was the part I really had trouble with in recursion-schemes the names of the functions for Foldable are… opaque if you’re not familiar with the terminology.

The most basic one is cata, which is the “catamorphism” across our data type. I’m not going to trouble you with why we call it a catamorphism, but just remember that it’s the souped-up version of foldr.

    foldr :: (a -> b -> b)          -> b -> [a]    -> b
    foldr :: ((a, b) -> b)          -> b -> [a]    -> b
    cata  :: (Fix BList a -> b)     -> List a      -> b
    cata  :: (Base (List a) a -> b) -> List a      -> b
    cata  :: (Base t b -> b)        -> t           -> b

And we can use it the same way!

    map :: (a -> b) -> List a -> List b
    map f = cata mapper
      where mapper (BCons a b) = f a `MyCons` b
            mapper BNil        = MyNil

    myfilter :: (a -> Bool) -> List a -> List a
    myfilter p = cata filterer
      where filterer (BCons a b) = if p a then a `MyCons` b else b
            filterer BNil        = MyNil

Now we can all tell people that we’ve written map using a catamorphism.

Careful readers will notice one big difference between foldr and cata: cata doesn’t take a seed! Indeed with foldr we replace all the constructors of our list with the function f, so

    1 : 2 : 3 : 4 : []
    1 `f` 2 `f` 3 `f` 4 `f` seed

This doesn’t generalize well though, what if we have a type with a constructor of 3 arguments? Or 5? To avoid this problem, recursion-schemes takes a clever approach.

Remember that BList factors out recursion? cata works by collapsing a sublist recursively and sticking the slot back into the slot of the original list. So we actually have something like

    BCons 1 (BCons 2 (BCons 3 (BCons 4 BNil)))
    BCons 1 (f (BCons 2 (f (BCons 3 (f (BCons 4 (f Nil)))))))

Now f has to handle all possible cases of our constructor, so it handles both the seed value and the collapsing case! And this generalizing beautifully by just delegating all the constructor specific work to f this is how it’s possible to derive cata practically for free.

Now, since recursion-schemes already has an instance for [a], I’ll dispense with MyList since it’s a bit clunky.

Our foldable instance gives us quite a bit more than just foldr however! We also get this function para, short for “paramorphisms”. A paramorphism is like a fold, but also gives a “snapshot” of the structure at the point we’re folding. So if we wanted to sum each tail of a list, we could do something like

    sumTails :: Num a => [a] -> [a]
    sumTails = para summer
      where summer (Cons a (list, rest)) = a + sum list : rest
            summer Nil                   = []

This could be useful for example, if you’re doing any context dependent operations on a structure. Later, I’ll try to include some more practical examples of a paramorphism (I never thought I’d say those words).

Now recursion-schemes includes generalized versions of all of these but I’m not brave enough to try to explain them right now.

A Real Example

Before we wrap this post up, let’s demonstrate an actual useful example of recursion-schemes.

We’re going to implement trivial constant folding in a made up language I’ll call Foo.

The AST for Foo is something like

    data Op = Plus | Sub | Mult | Div

    data Foo = Num Int           -- Numeric literals
             | String String     -- String literals
             | Binop Op Foo Foo  -- Primitive operation
             | Fun String Foo    -- Lambda/Abstraction over terms
             | App Foo Foo       -- Application
             | Var String        -- Variables
             deriving Show

Now we want our trivial constant folding to reduce something like Binop Plus (Num 1) (Num 2) to just Num 3. Let’s first formalize this by writing a quick little reducer

    compute :: Op -> Int -> Int -> Int
    compute Plus = (+)
    compute Sub  = (-)
    compute Mult = (*)
    compute Div  = div

    reduce :: Foo -> Foo
    reduce (Binop op (Num a) (Num b)) = Num $ compute op a b -- The reduction
    reduce a                          = a

So we compute all constant expressions and leave everything else alone. This is pretty simple, but how can we apply it to every element in our AST? Well, time to break out recursion-schemes

    data FooB a = NumB Int
                | StringB String
                | BinopB Op a a
                | FunB String a
                | App a a
                | Var String
    type instance Base Foo = FooB

    instance Foldable Foo where
      project (Num a)        = NumB a
      project (String a)     = StringB a
      project (Binop op a b) = BinopB op a b
      project (Fun v a)      = FunB v a
      project (App a b)      = AppB a b
      project (Var a)        = VarB a

    -- reverse of project
    rProject :: Base Foo Foo -> Foo
    rProject (NumB a)        = Num a
    rProject (StringB a)     = String a
    rProject (BinopB op a b) = Binop op a b
    rProject (FunB v a)      = Fun v a
    rProject (AppB a b)      = App a b
    rProject (VarB a)        = Var a

And let’s rewrite reduce to use FooB instead of Foo

    reduce :: Base Foo Foo -> Foo
    reduce (Fix (BinopB op (Num a) (Num b))) = Num $ compute op a b -- The reduction
    reduce a                                 = rProject a

So this entire traversal now just becomes

    constFold :: Foo -> Foo
    constFold = cata reduce

Now we can test our simple optimization

    test = Binop Plus (Num 1) (Binop Mult (Num 2) (Num 3))
    optimized = constFold test
    main = print optimized

As we’d hope, this prints out Num 7!

This seems like a lot of work but don’t forget, now that we’ve taught recursion-schemes how to do traversals, we get all of this for free. For example, let’s now write a function to grab all the free variables of an expression.

As before, let’s start by writing the simple worker function for this traversal.

     freeVar :: Base Foo [String] -> [String]
     freeVar (NumB _)         = []
     freeVar (StringB _)      = []
     freeVar (VarB s)         = [s]
     freeVar (BinopB _ v1 v2) = v1 ++ v2
     freeVar (AppB v1 v2)     = v1 ++ v2
     freeVar (FunB v vs)      = delete v vs

Now the full traversal is trivial!

    freeIn :: Foo -> [String]
    freeIn = cata freeVars

As we’d hope, this traversal is much easier to write than the first one. You can imagine that the boilerplate of writing FooB and project is amortized over each traversal, making it much easier to write subsequent traversals once we’ve gone through the trouble of actually laying down the foundation.

What’s Next?

So far I’ve discussed part of the Foldable half of the recursion-schemes library. In my next post I’ll cover anamorphisms and Unfoldable, the dual of what we’ve talked about here.

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