Anamorphism

For other uses, see Anamorphosis (disambiguation).

In functional programming, an anamorphism is a kind of generic function that can corecursively construct a result of a certain type and which is parameterized by functions that determine what the next single step of the construction is. The term is composed of the prefix ana (from Greek ἀνά = upwards) + the stem morphism (from Greek μορφή = form, shape).

The concept of anamorphism, which has its grounding in category theory, generalizes the list-producing unfold functions to arbitrary algebraic data types that can be described as final coalgebras. In fact, the terms 'anamorphism' and "unfold" (as a noun) are often used interchangeably. Anamorphisms, which are co-recursive, are dual to catamorphisms, which are recursive, just as unfolds are dual to folds.

Examples

Anamorphisms on lists

An unfold on lists would build a (potentially infinite) list from a seed value. Typically, the unfold takes a seed value x, a one-place operation unspool that yields a pairs of such items, and a predicate finished which determines when to finish the list (if ever). In the action of unfold, the first application of unspool, to the seed x, would yield unspool x = (y,z). The list defined by the unfold would then begin with y and be followed with the (potentially infinite) list that unfolds from the second term, z, with the same operations. So if unspool z = (u,v), then the list will begin y:u:..., where ... is the result of unfolding v with r, and so on.

A Haskell definition of an unfold, or anamorphism for lists, called ana, is as follows:

-- The type signature of 'ana':
ana :: (b->(a,b)) -> (b->Bool)-> b -> [a]   
-- Its definition:
ana unspool finished x = if finished x
                        then []
                        else  a : ana unspool finished y
                            where (a,y) = unspool x 

(Here finished and unspool are parameters of the function, like x.) We can now implement quite general functions using ana.

Anamorphisms on other data structures

An anamorphism can be defined for any recursive type, according to a generic pattern. For example, the unfold for the tree data structure

data Tree a = Leaf a | Branch (Tree a) a (Tree a)

is as follows

ana :: (b->Either a (b,a,b)) -> b -> Tree a
ana unspool x = case unspool x of
                  Left a -> Leaf a
                  Right (l,x,r) -> Branch (ana unspool l) x (ana unspool r)

History

One of the first publications to introduce the notion of an anamorphism in the context of programming was the paper Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire, by Erik Meijer et al., which was in the context of the Squiggol programming language.

Applications

Functions like zip and iterate are examples of anamorphisms. zip takes a pair of lists, say ['a','b','c'] and [1,2,3] and returns a list of pairs [('a',1),('b',2),('c',3)]. Iterate takes a thing, x, and a function, f, from such things to such things, and returns the infinite list that comes from repeated application of f, i.e. the list [x, (f x), (f (f x)), (f (f (f x))), ...].

zip (a:as) (b:bs) = if (as==[]) || (bs ==[])   -- || means 'or'
                     then [(a,b)]
                     else (a,b):(zip as bs) 

iterate f x = x:(iterate f (f x)) 

To prove this, we can implement both using our generic unfold, ana, using a simple recursive routine:

zip2 = ana g p
   where
   p (as,bs) = (as==[]) || (bs ==[]) 
   g ((a:as), (b:bs)) = ((a,b),(as,bs))

iterate2 f = ana (\a->(a,f a)) (\x->False) 

In a language like Haskell, even the abstract functions fold, unfold and ana are merely defined terms, as we have seen from the definitions given above.

Anamorphisms in category theory

In category theory, anamorphisms are the categorical dual of catamorphisms (and catamorphisms are the categorical dual of anamorphisms).

That means the following. Suppose (A, fin) is a final F-coalgebra for some endofunctor F of some category into itself. Thus, fin is a morphism from A to FA, and since it is assumed to be final we know that whenever (X, f) is another F-coalgebra (a morphism f from X to FX), there will be a unique homomorphism h from (X, f) to (A, fin), that is a morphism h from X to A such that fin . h = Fh . f. Then for each such f we denote by ana f that uniquely specified morphism h.

In other words, we have the following defining relationship, given some fixed F, A, and fin as above:

• $h = \mathrm{ana}\ f$
• $fin\circ h = Fh \circ f$

Notation

A notation for ana f found in the literature is $[\!(f)\!]$. The brackets used are known as lens brackets, after which anamorphisms are sometimes referred to as lenses.

See also

• Catamorphism
• Hylomorphism
• Paramorphism
• Apomorphism

External links

• Anamorphisms in Haskell