Curse explicit recursion!

dev* 2018


What is recursion?

Yeah, old joke...

What is recursion?

Recursion in computer science is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem (as opposed to iteration). The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. 🤔

Factorial with iteration, in Java

int factorial(int n) {
    int res = 1;
    for (int i=1; i<=n; i++) {
        res = res * i;
    return res;

With mutation included, as iteration often does.

Standard response of functional languages: recursion!

factorial 0 = 1
factorial n = n * factorial (n-1)

Maybe elegant but there's a problem...

It's not tail recursive -> will blow up stack usage

Recursive list length

(define (length lst)
        [(empty? lst) 0]
        [else (+ 1 (length lst))]))

But there’s a bug not easily seen!

This is in Scheme, so is this a...

( •_•)>⌐■-■

... Recursion Scheme?

Recursion is dangerous!

Do we really need recursion?

factorial n = foldr (*) 1 [1..n]

No recursion to be seen! But what is this “fold” thing?

  • Fold hides the recursion in a safe wrapping
  • Fold is actually something called a catamorphism
  • There are other wrappers. Ever heard of an unfold?

more Haskell factorials:

Can we “package” all kinds of recursion?

Could we even have a usable programming language without any kind of explicit recursive calls?

I have no idea 🙂

But why?

  • separatation of concerns
  • separate actual algorithm from the recursive part
    • e.g. for length the "business logic” is:
      1 + length_of_rest_of_the_list
  • cleaner!
  • generally usable with different structures (trees?)

Recursion Schemes

Brief and incomplete history

Is this still pop?

Is this still pop?

Implementing some use cases

For these examples we need something called
a fix point

newtype Fix f = Fix (f (Fix f))

-- Wrap value inside a Fix point
fix :: f (Fix f) -> Fix f
fix = Fix

-- Unwrap value from a Fix point
unfix :: Fix f -> f (Fix f)
unfix (Fix f) = f

Learning about it is left as an exercise for the reader 😉

Example 1: sum of a list of numbers (catamorphism)

-- Familiar cons-list. No need to make this, but as an example...
data MyListF elem rest = NilF | ConsF elem rest deriving Functor
type List t = Fix (MyListF t)

nil = Fix NilF -- make an empty list
cons x xs = Fix (ConsF x xs) -- cons a value in front of a list

-- our "algorithm" to sum list elements would be just:
sumAlg = \case
    NilF       -> 0      -- sum of an empty list is 0
    ConsF x xs -> x + xs -- add the current value to others

-- execute our algorithm by giving it to some library function
exampleSum = doStuff sumAlg
-- > exampleSum (cons 4 $ cons 12 $ cons 7 nil) -- [4, 12, 7]
-- 23

Example 1: sum of a list of numbers (catamorphism)

-- This general library function still needs explicit recursion
doStuff alg = unfix >>>          -- unwrap from the Fix point
              fmap (doStuff alg) -- apply algorithm recursively
              >>> alg            -- apply to the final value

-- "alg" was not an abreviation of "algorithm", but of "Algebra"
type Algebra f a = f a -> a

-- Complete implementation for 'sum' would thus be the
-- following (using recursion-schemes library), which neatly
-- contains only the algorithmic part without any recursion
sum :: [Int] -> Int
sum = cata $ \case
    Nil       -> 0
    Cons x xs -> x + xs

Sum is just a regular fold, so the famous fold is just one of the recursion schemes, namely catamorphism!

Reminder: Many things come in pairs

Example 2: natural numbers (anamorphism)

-- Stream of natural numbers
data StreamF r = StreamF Natural r deriving Functor
type NatStream = Fix StreamF

-- our "algorithm" to create numbers would be
naturalsCoalg n = StreamF n (n + 1) -- for each n, next is n+1

-- same as before, seed value zero
exampleNats = doStuff naturalsCoalg 0

-- unfortunately in this example our numbers are our NatStream,
-- but we can convert it to a Haskell list with a catamorphism
exampleNaturals = cata (\case StreamF x xs -> x:xs) exampleNats
-- > take 4 exampleNaturals
-- [0,1,2,3]

Example 2: natural numbers (anamorphism)

-- Dual of catamorphism:
-- 1) arrows are reversed
-- 2) wrap instead of unwrap
doStuff coalg = Fix <<< fmap (doStuff coalg) <<< coalg

-- "coalg" is called a "Coalgebra"
type Coalgebra f a = a -> f a

-- Complete implementation using recursion-schemes library:
naturals :: [Natural]
naturals = ana (\n -> Cons n (n + 1)) 0

So, generating an infinite stream is just one of the recursion schemes, namely anamorphism!

Sidetrack: fan out operator

in Data.Tuple.Extra
or Control.Arrow
or Control.Category.Cartesian

-- for functions:
(&&&) :: (a -> b) -> (a -> c) -> a -> (b, c)
-- for any category:
(&&&) :: a b c -> a b c' -> a b (c, c')

"split a value to two different paths”

Example 3: factorial (paramorphism)

-- Natural numbers as Peano
data NatF r = ZeroF | SuccF r deriving Functor
type Nat = Fix NatF

zero = Fix ZeroF
succ = Fix . SuccF

factorialAlg = \case
    ZeroF          -> 1
    SuccF (cur, r) -> (toNat cur + 1) * r
    -- unfortunately our example numbers are Peano, but we
    -- can use catamorphism to transform to regular numbers
  where toNat = cata $ \case
        ZeroF   -> 0
        SuccF n -> n + 1

exampleFactorial = doStuff factorialAlg
-- > exampleFactorial (succ (succ (succ zero))) -- 3
-- 6

Example 3: factorial (paramorphism)

doStuff alg = unfix >>> fmap (id &&& doStuff alg) >>> alg

-- here "alg" is called an "RAlgebra"
type RAlgebra f a = f (Fix f, a) -> a

-- Complete implementation using recursion-schemes library:
factorial :: Natural -> Natural
factorial = para $ \case
    Nothing     -> 1
    Just (n, r) -> (n+1) * rx

Thus, catamorphism with access to the current input value is called a paramorphism!

Sidetrack: fan in operator

dual for fan out

in Control.Arrow
or Control.Category.Cartesian

(|||) :: a b d -> a c d -> a (Either b c) d 

“split a valut to one of two different paths”

Example 4: numeric range (apomorphism)

rangeCoalg to = \case
    n | n == to -> ConsF n (Left nil)
    n           -> ConsF n (Right (n + 1))

exampleRange_ from to = doStuff (rangeCoalg to) from
-- have to convert our List type to the Haskell list again...
exampleRange from to = cata (\case
    NilF       -> []
    ConsF x xs -> x:xs
) (exampleRange_ from to)
-- > exampleRange 0 4
-- [0,1,2,3,4]

Example 4: numeric range (apomorphism)

doStuff coalg = Fix <<< fmap (id ||| doStuff coalg) <<< coalg

-- here "coalg" is called an "RCoalgebra":
type RCoalgebra f a = a -> f (Either (Fix f) a)

-- Complete implementation using recursion-schemes library:
range :: Int -> Int -> [Int]
range from to = flip apo from $ \case
    n | n == to -> Cons n (Left [])
    n           -> Cons n (Right (n + 1))

That is: anamorphism terminating at some chosen point, is called an apomorphism!

Sidetrack: Freedom!

Free Monad:

data Free f a = Pure a | Free (f (Free f a))

Cofree Comonad:

data Cofree f a = a :< (f (Cofree f a))

Kind of minimal way to provide monadic or comonadic structure

Example 5: fibonacci (histomorphism)

Sometimes we need access to the previous values

-- using Cofree Comonad here to bring comonadic structure...
fibonacciAlg ZeroF                         = 0
fibonacciAlg (SuccF (_ :< ZeroF))          = 1
fibonacciAlg (SuccF (a :< SuccF (b :< _))) = a + b

exampleFibonacci = doStuff fibonacciAlg
-- > exampleFibonacci (succ (succ (succ zero))) -- 3
-- 2

Example 5: fibonacci (histomorphism)

doStuff alg = f >>> (\(x :< _) -> x)
    where f = unfix >>> fmap f >>> (alg &&& id) >>> uncurry (:<)

-- here "alg" is called a "CVAlgebra"
type CVAlgebra f a = f (Cofree f a) -> a

-- Complete implementation using recursion-schemes library:
fibonacci :: Natural -> Natural
fibonacci = histo $ \case
    Nothing                   -> 0
    Just (_ :< Nothing)       -> 1
    Just (a :< Just (b :< _)) -> a + b

Paramorphism providing the whole history instead of the current value is called a histomorphism!

Example 6: capitalizeWords (futumorphism)

Apomorphism let us terminate once, but now we need to terminate multiple times with a value (word)

capitalizeWordsCoalg ""     = NilF
capitalizeWordsCoalg str = let 
    ws = words str
    (first,rest) = (head ws, tail ws)
    capitalized = toUpper (head first) : tail first
  in ConsF capitalized (Pure $ unwords rest)

exampleCapitalizeWords_ = doStuff capitalizeWordsCoalg
exampleCapitalizeWords str = unwords $ cata (\case
    NilF       -> []
    ConsF x xs -> x:xs
  ) (exampleCapitalizeWords_ str)
-- > exampleCapitalizeWords "foo bar"
-- "Foo Bar"

Example 6: capitalizeWords (futumorphism)

doStuff coalg = Fix <<< fmap worker <<< coalg
where worker (Pure a) = doStuff coalg a
      worker (Free a) = Fix (fmap worker a)

-- here "coalg" is called a "CVCoalgebra":
type CVCoalgebra f a = a -> f (Free f a)

-- Complete implementation using recursion-schemes library:
capitalizeWords :: String -> String
capitalizeWords = unwords . futu (\case
    ""  -> Nil
    str -> let ws = words str
               (first,rest) = (head ws, tail ws)
               capitalized = toUpper (head first) : tail first
           in Cons capitalized (Pure $ unwords rest))

Whereas histomorphism extended paramorphism to the whole history, futumorphism extends apomorphism to the whole future!

Example 7: takeWhile (prepromorphism)

-- filtering of a List-functor with a predicate function
filter _ Nil = Nil
filter pred x@(Cons h _)
    | pred h    = x
    | otherwise = Nil

takeWhile :: (a -> Bool) -> [a] -> [a]
takeWhile pred = prepro (filter pred) $ \case
    Nil         -> []
    (Cons x xs) -> x:xs

Prepromorphism is like a catamorphism but executes a natural transformation before the actual stuff

Example 8: first 10 natural numbers (postpromorphism)

With Apomorphism we could terminate the recursion, but it would be nicer if we could work on infinite streams and cap them with an external function.

-- use whatever generator function
generate f n = Cons n (f n)

-- now both filtering and stream generation are nicely separated from recursion
range :: (Num a, Ord a) => a -> a -> [a]
range from to = postpro (filter (<= to)) (generate (+1)) from

Postpromorphism is like an anamorphism but executes a natural transformation after the actual stuff


Many times a problem involves multiple recursions.

For example: Merge sort

  1. unfold with anamorphism
  2. fold with catamorphism

using ana + cata is called a hylomorphism


hylo :: Functor f => Algebra f b -> Coalgebra f a -> a -> b  
hylo alg coalg = ana coalg >>> cata alg

Can be implemented without actually invoking cata or ana.

Also it seems that the compiler can optimize away all our imaginary structure used to define Algebra and Coalgebra! *



One can also combine Futumorphism + Histomorphism:

chrono :: Functor f => CVAlgebra f b -> CVCoalgebra f a -> a -> b  
chrono cvalg cvcoalg = futu cvcoalg >>> histo cvalg

Elgot algebras

Sometimes we need to short circuit our hylomorphism during construction

elgot :: Functor f => Algebra f b -> (a -> Either b (f a)) -> a -> b

or during destruction

coelgot :: Functor f => ((a, f b) -> b) -> Coalgebra f a -> a -> b  


Sorting algorithms tend to be different combinations of specific recursion patterns.

Many (if not all?) sorting algorithms can be defined elegantly with suitable recursion schemes.


data vs codata


Different algebras deal with data. They provide recursion.

Different coalgebras deal with codata. They provide corecursion.

Some functions are total for data but not for codata (e.g. length).

Some functions are total for codata but not for data (e.g. head).

data vs codata

In a total language (not Haskell!) the difference is important, since all functions must always terminate and never fail.

If our language separated data and codata,
could we have safe recursion for data and safe corecursion for codata?!?


catamorphism“romauttaminen”type Algebra f a = f a -> a
anamorphism“hajauttaminen”type Coalgebra f a = a -> f a
paramorphism“liittäminen”type RAlgebra f a = f (Fix f, a) -> a
apomorphism“erottaminen”type RCoalgebra f a = a -> f (Either (Fix f) a)
histomorphism“historia”type CVAlgebra f a = f (Cofree f a) -> a
futumorphism“tulevaisuus”type CVCoalgebra f a = a -> f (Free f a)
hylomorphism“hajauttaminen + romauttaminen”
chronomorphism“tulevaisuus + historia”

zygomorphism, dynamorphism, codynamorphism, elgot algebra, elgot coalgebra, metamorphism, synchromorphism, exomorphism, mutumorphism, ...


Reading material for passionate souls

Cheat sheet
First mention of catamorphism
Data structures and program transformation
Algebraic Data Types and Program Transformation
Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire
Primitive (Co)Recursion and Course-of-Value (Co)Iteration, Categorically (Uustalu, Vene)
Great blog series, which I can understand
More readable blog posts
Table of recursion schemes
Elgot algebras
Edward Kmett’s Haskell library
About data vs codata
A curation of useful resources for learning about and using recursion schemes
About catamorphisms

Remember to learn
a category theoretical thing
every year ;)

Thank you!