# recursion-schemes

Representing common recursion patterns as higher-order functions

http://github.com/ekmett/recursion-schemes/

LTS Haskell 18.0: | 5.2.2.1 |

Stackage Nightly 2021-06-20: | 5.2.2.1 |

Latest on Hackage: | 5.2.2.1 |

**Edward A. Kmett**

`recursion-schemes-5.2.2.1@sha256:fed7167e83698147d7c078bbb1fe74451f96c7c55ed9986c734d268070d85864,3728`

#### Module documentation for 5.2.2.1

- Data
- Data.Functor

*(full list with versions)*:

# 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
| Add Expr Expr
| 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.

## Contributing

Contributions and bug reports are welcome!

## Changes

## 5.2.2.1

- Fix build issue regarding
`Setup.hs`

. See #120.

## 5.2.2

- More Mendler-style recursion-schemes:
`mpara`

,`mzygo`

,`mana`

,`mapo`

, and`mfutu`

. `makeBaseFunctor`

no longer generates warnings when combined with DerivingStrategies.

## 5.2.1 [2020.10.04]

- Allow building with
`template-haskell-2.17.0.0`

(GHC 9.0).

## 5.2

- Add instances for
`Tree`

(from`containers`

) - Add some haddocks and basic examples
- Generalize the type of
`makeBaseFunctor(With)`

, such that it can take also`Dec`

. This way you may supply context for`Recursive`

and`Corecursive`

instances. - Depend on
`data-fix`

package for fixed point types.

## 5.1.3 [2019.04.26]

- Support
`th-abstraction-0.3.0.0`

or later.

## 5.1.2

- Make the
`Generic`

-based instances to also support data constructors with zero arguments (and datatypes with zero constructors).

## 5.1.1.1

- Invalid release

## 5.1.1

- Add
`cotransverse`

- Add
`Generic`

based default implementation to`embed`

and`project`

.`Recursive`

and`Corecursive`

can be`DeriveAnyClass`

-derived now, if you write the base functor by hand.

## 5.1

- Export gfutu
`distGHisto`

,`ghisto`

, and`gchrono`

now use`Cofree (Base t)`

`distGFutu`

,`gfutu`

, and`gchrono`

now use`Free (Base t)`

- Add
`hoist`

,`hoistMu`

and`hoistNu`

- Add
`transverse`

and`cataA`

## 5.0.3 [2018.07.01]

- Make the Template Haskell machinery look through type synonyms.
- Avoid incurring some dependencies when using recent GHCs.

## 5.0.2

- Support GHC-8.2.1
- Fix Template Haskell derivation with non-default type renamer.
- Add
`Recursive`

and`Corecursive Natural`

instances, with`Base Natural = Maybe`

.

## 5.0.1

- Add
`Data.Functor.Foldable.TH`

module, which provides derivation of base functors via Template Haskell.

## 5

- Renamed
`Foldable`

to`Recursive`

and`Unfoldable`

to`Corecursive`

. With`Foldable`

in`Prelude`

in GHC 7.10+, having a needlessly conflicting name seemed silly. - Add support for GHC-8.0.1
- Use
`Eq1`

,`Ord1`

,`Show1`

,`Read1`

to derive`Fix`

,`Nu`

and`Mu`

`Eq`

,`Ord`

`Show`

and`Read`

instances - Remove
`Prim`

data family.`ListF`

as a new name for`Prim [a]`

, with plenty of instances, e.g.`Traversable`

. - Export
`unfix`

- Add chronomorphisms:
`chrono`

and`gchrono`

. - Add
`distGApoT`

## 4.1.2

- Support for
`free`

4.12.1

## 4.1.1

- Support for GHC 7.10
- Fixed
`para`

.

## 4.1

- Support for GHC 7.7+’s generalized
`Typeable`

. - Faster
`gapo`

and`para`

by exploiting sharing.

## 4.0

- Compatibility with
`comonad`

and`free`

version 4.0

## 3.0

- Compatibility with
`transformers`

0.3 - Resolved deprecation warnings caused by changes to
`Data.Typeable`