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Concept
Anamorphism
Summary
In computer programming, an anamorphism is a function that generates a sequence by repeated application of the function to its previous result. You begin with some value A and apply a function f to it to get B. Then you apply f to B to get C, and so on until some terminating condition is reached. The anamorphism is the function that generates the list of A, B, C, etc. You can think of the anamorphism as unfolding the initial value into a sequence.
The above layman's description can be stated more formally in : the anamorphism of a coinductive type denotes the assignment of a coalgebra to its unique morphism to the final coalgebra of an endofunctor. These objects are used in functional programming as unfolds.
The categorical dual (aka opposite) of the anamorphism is the catamorphism.
In functional programming, an anamorphism is a generalization of the concept of unfolds on coinductive lists. Formally, anamorphisms are generic functions that can corecursively construct a result of a certain type and which is parameterized by functions that determine the next single step of the construction.
The data type in question is defined as the greatest fixed point ν X . F X of a functor F. By the universal property of final coalgebras, there is a unique coalgebra morphism A → ν X . F X for any other F-coalgebra a : A → F A. Thus, one can define functions from a type A into a coinductive datatype by specifying a coalgebra structure a on A.
As an example, the type of potentially infinite lists (with elements of a fixed type value) is given as the fixed point [value] = ν X . value × X + 1, i.e. a list consists either of a value and a further list, or it is empty. A (pseudo-)Haskell-Definition might look like this:
data [value] = (value:[value]) | []
It is the fixed point of the functor F value, where:
data Maybe a = Just a | Nothing
data F value x = Maybe (value, x)
One can easily check that indeed the type [value] is isomorphic to F value [value], and thus [value] is the fixed point.
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