Catamorphisms

Folding data structures

An overview and derivation of the category-theoretic notion of a catamorphism as a recursion scheme, and an exploration of common variations on the theme.

Description

Catamorphisms are generalizations of the concept of a fold in functional programming. A catamorphism deconstructs a data structure with an F-algebra for its underlying functor.

History

The name catamorphism appears to have been chosen by Lambert Meertens [1]. The category theoretic machinery behind these was resolved by Grant Malcolm [2][3], and they were popularized by Meijer, Fokkinga and Paterson[4][5]. The name comes from the Greek 'κατα-' meaning "downward or according to". A useful mnemonic is to think of a catastrophe destroying something.

Notation

A catamorphism for some F-algebra (X,f) is denoted (| f |)F. When the functor F can be determined unambiguously, it is usually written (|φ|) or cata φ. Due to this choice of notation, a catamorphism is sometimes called a banana and the (|.|) notation is sometimes referred to as banana brackets.

Haskell Implementation

type Algebra f a = f a -> a 
newtype Mu f = InF { outF :: f (Mu f) } 
cata :: Functor f => Algebra f a -> Mu f -> a 
cata f = f . fmap (cata f) . outF

Alternate Definitions

cata f = hylo f outF 
cata f = para (f . fmap fst)

Duality

A catamorphism is the categorical dual of an anamorphism.

Derivation

If (μF,inF) is the initial F-algebra for some endofunctor F and (X,φ) is an F-algebra, then there is a unique F-algebra homomorphism from (μF,inF) to (X,φ), which we denote (| φ |)F.

That is to say, the following diagram commutes:

Laws

Rule	Haskell
cata-cancel	cata phi . InF = phi . fmap (cata phi)
cata-refl	cata InF = id
cata-fusion	f . phi = psi . fmap f implies f . cata phi = cata psi
cata-compose	cata phi . cata (InF . eps) = cata (phi . eps) if eps :: f ~> g is a natural transformation

Examples

The underlying functor for a string of Chars and its fixed point

  data StrF x = Cons Char x | Nil 
  type Str = Mu StrF
  instance Functor StrF where     
    fmap f (Cons a as) = Cons a (f as)     
    fmap f Nil = Nil

The length of a string as a catamorphism.

  length :: Str -> Int 
  length = cata phi where     
    phi (Cons a b) = 1 + b     
    phi Nil = 0

The underlying functor for the natural numbers.

  data NatF a = S a | Z deriving (Eq,Show) 
  type Nat = Mu NatF  
  instance Functor NatF where      
    fmap f Z = Z      
    fmap f (S z) = S (f z)

Addition as a catamorphism.

  plus :: Nat -> Nat -> Nat 
  plus n = cata phi where      
    phi Z = n      
    phi (S m) = s m

Multiplication as a catamorphism

  times :: Nat -> Nat -> Nat 
  times n = cata phi where      
    phi Z = z      
    phi (S m) = plus n m  
  z :: Nat 
  z = InF Z  
  s :: Nat -> Nat 
  s = InF . S

Mendler Style

A somewhat less common variation on the theme of a catamorphism is a catamorphism as a recursion scheme a la Mendler, which removes the dependency on the underlying type being an instance of Haskell's Functor typeclass [6].

  type MendlerAlgebra f c = forall a. (a -> c) -> f a -> c [8]
  mcata :: MendlerAlgebra f c -> Mu f -> c 
  mcata phi = phi (mcata phi) . outF

From which we can derive the original notion of a catamorphism:

  cata :: Functor f => Algebra f c -> Mu f -> c 
  cata phi = mcata (\f -> phi . fmap f)

This can be seen to be equivalent to the original definition of cata by expanding the definition of mcata.

The principal advantage of using Mendler-style is it is independent of the definition of the Functor definition for f.

Mendler and the Contravariant Yoneda Lemma

The definition of a Mendler-style algebra above can be seen as the application of the contravariant version of the Yoneda lemma to the functor in question.

In type theoretic terms, the contravariant Yoneda lemma states that there is an isomorphism between (f a) and ∃b. (b -> a, f b), which can be witnessed by the following definitions.

  data CoYoneda f a = forall b. CoYoneda (b -> a) (f b)  
  toCoYoneda :: f a -> CoYoneda f a 
  toCoYoneda = CoYoneda id  
  fromCoYoneda :: Functor f => CoYoneda f a -> f a 
  fromCoYoneda (CoYoneda f v) = fmap  f v

Note that in Haskell using an existential requires the use of data, so there is an extra bottom that can inhabit this type that prevents this from being a true isomorphism.

However, when used in the context of a (CoYoneda f)-Algebra, we can rewrite this to use universal quantification because the functor f only occurs in negative position, eliminating the spurious bottom.

  Algebra (CoYoneda f) a 
    = (by definition) CoYoneda f a -> a 
    ~ (by definition) (exists b. (b -> a, f b)) -> a 
    ~ (lifting the existential) forall b. (b -> a, f b) -> a 
    ~ (by currying) forall b. (b -> a) -> f b -> a 
    = (by definition) MendlerAlgebra f a

Generalized Catamorphisms

Most more advanced recursion schemes for folding structures, such as paramorphisms and zygomorphisms can be seen in a common framework as "generalized" catamorphisms[7]. A generalized catamorphism is defined in terms of an F-W-algebra and a distributive law for the comonad W over the functor F which preserves the structure of the comonad W.

  type Dist f w = forall a. f (w a) -> w (f a) 
  type FWAlgebra f w a = f (w a) -> a 
  g_cata :: (Functor f, Comonad w) => 
          Dist f w -> FWAlgebra f w a -> Mu f -> a 
  g_cata k g = extract . c where 
    c = liftW g . k . fmap (duplicate . c) . outF

However, a generalized catamorphism can be shown to add no more expressive power to the concept of a catamorphism. That said the separation of a number of the "book keeping" concerns by isolating them in a reusable distributive law can ease the development of F-W-algebras.

We can transform an F-W-algebra into an F-algebra by including the comonad in the carrier for the algebra and then extracting after we perform this somewhat more stylized catamorphism:

  lowerAlgebra :: (Functor f, Comonad w) => 
                Dist f w -> FWAlgebra f w a -> Algebra f (w a) 
  lowerAlgebra k phi = liftW phi . k . fmap duplicate 
  g_cata :: (Functor f, Comonad w) => 
          Dist f w -> FWAlgebra f w a -> Mu f -> a 
  g_cata k phi = extract . cata (lowerGAlgebra k phi)

and we can trivially transform an Algebra into an F-W-Algebra by mapping the counit of the comonad over F. Then using the trivial identity functor, we can represent every catamorphism as a generalized-catamorphism.

  liftAlgebra :: (Functor f, Comonad w) => 
    Algebra f a -> FWAlgebra f w a

  liftAlgebra phi = phi . fmap extract


  cata :: Functor f => Algebra f a -> Mu f -> a
  cata f = g_cata (Identity . fmap runIdentity) (liftAlgebra f)

Between these two definitions we can see that a generalized catamorphism does not increase the scope of a catamorphism to encompass any more operations, it simply further stylizes the pattern of recursion.

References

1. L. Meertens. First Steps towards the theory of Rose Trees. Draft Report, CWI, Amsterdam, 1987.
2. G. Malcolm. PhD. Thesis. University of Gronigen, 1990.
3. G. Malcolm. Data structures and program transformation. Science of Computer Programming, 14:255--279, 1990.
4. E. Meijer. Calculating Compilers, Ph.D Thesis, Utrecht State University, 1992.
5. E. Meijer, M. Fokkinga, R. Paterson, Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire, 5th ACM Conference on Functional Programming Languages and Computer Architecture.
6. T. Uustalu, V. Vene. Coding Recursion a la Mendler. Proceedings 2nd Workshop on Generic Programming, WGP'2000, Ponte de Lima, Portugal, 6 July 2000
7. T. Uustalu, V. Vene, A. Pardo. Recursion schemes from Comonads. Nordic Journal of Computing. Volume 8 , Issue 3 (Fall 2001). 366--390, 2001 ISSN:1236-6064
8. V. Vene. Categorical programming with inductive and coinductive types, PhD thesis, University of Tartu, 2000.
9. E. Kmett. Catamorphism. The Comonad.Reader, 2008.