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# recursion-schemes / recursion-schemes

Generalized bananas, lenses and barbed wire

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# recursion-schemes

This package represents common recursion patterns as higher-order functions.

## A familiar example

Here are two recursive functions.

```sum :: [Int] -> Int
sum [] = 0
sum (x:xs) = x + sum xs

product :: [Int] -> Int
product [] = 1
product (x:xs) = x * product xs```

These functions are very similar. In both cases, the empty list is the base case. In the cons case, each makes a recursive call on the tail of the list. Then, the head of the list is combined with the result using a binary function.

We can abstract over those similarities using a higher-order function, `foldr`:

```sum     = foldr (+) 0
product = foldr (*) 1```

## Other recursive types

`foldr` works great for lists. The higher-order functions provided by this library help with other recursive datatypes. Here are two recursive functions on `Tree`s:

```depth :: Tree a -> Int
depth (Node _ subTrees) = 1 + maximum subTrees

size :: Tree a -> Int
size (Node _ subTrees) = 1 + sum subTrees```

It is not possible to use `foldr` to simplify `depth`. Conceptually, `foldr` is flattening all the elements of the tree into a list before combining them with the binary function. This does not work for `depth` because it needs to examine the structure of the tree, which `foldr` flattened away.

We can instead use one of the higher-order functions provided by this library, `cata`.

```import Data.Functor.Base (TreeF(..))
import Data.Functor.Foldable

-- data Tree  a   = Node  a [Tree a]
-- data TreeF a r = NodeF a [r     ]

depth :: Tree a -> Int
depth = cata go
where
go :: TreeF a Int -> Int
go (NodeF _ subDepths) = 1 + maximum subDepths

size :: Tree a -> Int
size = cata go
where
go :: TreeF a Int -> Int
go (NodeF _ subSizes) = 1 + sum subSizes```

In this example, the code is a bit longer, but it is correct. Did you spot the mistake in the version which does not use `cata`? We forgot a call to `fmap`:

```depth :: Tree a -> Int
depth (Node _ subTrees) = 1 + maximum (fmap depth subTrees)

size :: Tree a -> Int
size (Node _ subTrees) = 1 + sum (fmap size subTrees)```

`cata` automatically adds this call to `fmap`. This is why `subDepths` contains a list of already-computed depths, not a list of sub-trees. In general, each recursive position is replaced by the result of a recursive call. These results have type `Int`, not type `Tree`, so we need a helper datatype `TreeF` to collect these results.

When you think about computing the depth, you probably think "it's 1 plus the maximum of the sub-depths". With `cata`, this is exactly what we write. By contrast, without `cata`, we need to describe both the "how" and the "what" in our implementation. The "how" is about recurring over the sub-trees (using `fmap depth`), while the "what" is about adding 1 to the maximum of the sub-trees. `cata` takes care of the recursion, so you can focus solely on the "what".

A recursion-scheme is a function like `cata` which implements a common recursion pattern. It is a higher-order recursive function which takes a non-recursive function as an argument. That non-recursive function describes the part which is unique to your calculation: the "what".

## Types with many constructors

Let's look at a more complex example. Here is a small lambda-calculus and a function to compute the free variables of an expression:

```import Data.Set (Set)
import qualified Data.Set as Set

data Expr
= Var String
| Lam String Expr
| App Expr Expr
| Constant Int
| Sub Expr Expr
| Mul Expr Expr
| Div Expr Expr
| ...

freeVars :: Expr -> Set String
freeVars (Var name)      = Set.singleton name
freeVars (Lam name body) = Set.difference (freeVars body) (Set.singleton name)
freeVars (App e1 e2)     = Set.union (freeVars e1) (freeVars e2)
freeVars (Constant _)    = Set.empty
freeVars (Add e1 e2)     = Set.union (freeVars e1) (freeVars e2)
freeVars (Sub e1 e2)     = Set.union (freeVars e1) (freeVars e2)
freeVars (Mul e1 e2)     = Set.union (freeVars e1) (freeVars e2)
freeVars (Div e1 e2)     = Set.union (freeVars e1) (freeVars e2)
freeVars ...```

As you can see, we had to repeat the `Set.union (freeVars e1) (freeVars e2)` line over and over. With recursion-schemes, this code becomes much shorter:

```{-# LANGUAGE DeriveFunctor, DeriveFoldable, DeriveTraversable, TemplateHaskell, TypeFamilies #-}
import Data.Functor.Foldable.TH (makeBaseFunctor)

makeBaseFunctor ''Expr

freeVars :: Expr -> Set String
freeVars = cata go
where
go :: ExprF (Set String) -> Set String
go (VarF name)           = Set.singleton name
go (LamF name bodyNames) = Set.difference bodyNames (Set.singleton name)
go fNames                = foldr Set.union Set.empty fNames```

The `makeBaseFunctor` line uses Template Haskell to generate our `ExprF` datatype, a single layer of the `Expr` datatype. `makeBaseFunctor` also generates instances which are useful when using recursion-schemes. For example, we make use of the `Foldable ExprF` instance on the last line of `go`. This `Foldable` instance exists because `ExprF` has kind `* -> *`, while `Expr` has kind `*`.

## Other recursion-schemes

All of our examples so far have used `cata`. There are many more recursion-schemes. Here is an example which follows a different recursive structure:

```-- |
-- >>> halves 256
-- [256,128,64,32,16,8,4,2,1]
halves :: Int -> [Int]
halves 0 = []
halves n = n : halves (n `div` 2)```

That recursive structure is captured by the `ana` recursion-scheme:

```halves :: Int -> [Int]
halves = ana go
where
go :: Int -> ListF Int Int
go 0 = Nil
go n = Cons n (n `div` 2)```

The Data.Functor.Foldable module provides many more.

## Flowchart for choosing a recursion-scheme

In addition to the choices described by the flowchart, you can always choose to use a refold.

## Contributing

Contributions and bug reports are welcome!

Generalized bananas, lenses and barbed wire