User:Ezrakilty/Draft of Anamorphism as generic function

In functional programming, an anamorphism is a kind of generic function which can (co)recursively construct a result of a certain type and which is parameterized by functions that determining what the next single step of the construction is. The term comes from Greek ἀνά (upwards) + morphism (from Greek μορφή, or form, shape) and the concept has its grounding in Category theory.

The concept of anamorphism 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

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Anamorphisms on Lists

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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

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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 Tree

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

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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,[1] by Erik Meijer et al., which was in the context of the Squiggol programming language.

Applications

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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 (fx)) 

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) 

Ruby comes with an explicit unfold operation; so, for example:

10.unfold { |n| n-1 unless n == 1 }.inspect 
       => [10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
10.class.unfold(&:superclass).inspect 
       => [Fixnum, Integer, Numeric, Object] 

In a language like Haskell, by contrast, 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

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

Notation

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A notation for ana f found in the literature is  . The brackets used are known as lens brackets, after which anamorphisms are sometimes referred to as lenses.

See also

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References

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  1. ^ Meijer, Erik (1991). "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire". {{cite web}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
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