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| In [[computer science]], and in particular [[functional programming]], a '''hylomorphism''' is a [[Recursion (computer science)|recursive]] function, corresponding to the [[function composition|composition]] of an [[anamorphism]] (which first builds a set of results; also known as 'unfolding') and a [[catamorphism]] (which then [[fold (higher-order function)|folds]] these results into a final [[return value]]). Fusion of these two recursive computations into a single recursive pattern then avoids building the intermediate data structure. This is a particular form of the optimizing program transformation techniques collectively known as [[deforestation (computer science)|deforestation]]. A related type of function is a '''metamorphism''', which is a catamorphism followed by an anamorphism.
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| ==Formal definition==
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| A hylomorphism <math>h : A \rightarrow C</math> can be defined in terms of its separate anamorphic and catamorphic parts.
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| The anamorphic part can be defined in terms of a [[arity|unary]] function <math>g : A \rightarrow B \times A</math> defining the list of elements in <math>B</math> by repeated application (''"unfolding"''), and a [[predicate (mathematics)|predicate]] <math>p : A \rightarrow \text{Boolean}</math> providing the terminating condition.
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| The catamorphic part can be defined as a combination of an initial value <math>c \in C</math> for the fold and a binary [[Operation (mathematics)|operator]] <math>\oplus : B \times C \rightarrow C</math> used to perform the fold.
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| Thus a hylomorphism
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| :<math>
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| h\,a = \begin{cases}
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| c & \mbox{if } p\,a
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| \\b \oplus ha' & \mbox{otherwise}
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| \end{cases}
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| </math>
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| (where <math>(b, a') = ga</math>) may be defined (assuming appropriate definitions of <math>p</math> & <math>g</math>).
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| ===Notation===
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| An abbreviated notation for the above hylomorphism is <math>h = [\![(c, \oplus),(g, p)]\!]</math>.
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| ==Hylomorphisms in practice==
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| ===Lists===
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| [[List (computing)|Lists]] are common data structures as they naturally reflect linear computational processes. These processes arise in repeated ([[Iteration|iterative]]) function calls. Therefore it is sometimes necessary to generate a temporary list of intermediate results before reducing this list to a single result.
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| One example of a commonly encountered hylomorphism is the canonical [[factorial]] function.
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| <source lang="haskell">
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| factorial :: Integer -> Integer
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| factorial n
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| | n == 0 = 1
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| | n > 0 = n * factorial (n - 1)
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| </source>
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| In the previous example (written in [[Haskell (programming language)|Haskell]], a [[purely functional]] [[programming language]]) it can be seen that this function, applied to any given valid input, will generate a linear call tree [[isomorphism|isomorphic]] to a list. For example, given ''n'' = 5 it will produce the following:
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| factorial 5 = 5 * (factorial 4) = 120
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| factorial 4 = 4 * (factorial 3) = 24
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| factorial 3 = 3 * (factorial 2) = 6
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| factorial 2 = 2 * (factorial 1) = 2
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| factorial 1 = 1 * (factorial 0) = 1
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| factorial 0 = 1
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| In this example, the anamorphic part of the process is the generation of the call tree which is isomorphic to the list <code>[1, 1, 2, 3, 4, 5]</code>. The catamorphism, then, is the calculation of the [[product (mathematics)|product]] of the [[element (mathematics)|elements]] of this list. Thus, in the notation given above, the factorial function may be written <math>\text{factorial} = [\![(1,\times),(g, p)]\!]</math> where <math>g\ n = (n, n - 1)</math> and <math>p\ n = (n = 0)</math>.
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| ===Trees===
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| However, the term 'hylomorphism' does not apply solely to functions acting upon isomorphisms of lists. For example, a hylomorphism may also be defined by generating a non-linear call tree which is then collapsed. An example of such a function is the function to generate the ''n''<sup>th</sup> [[term (mathematics)|term]] of the [[Fibonacci sequence]].
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| <source lang="haskell">
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| fibonacci :: Integer -> Integer
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| fibonacci n
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| | n == 0 = 0
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| | n == 1 = 1
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| | n > 1 = fibonacci (n - 2) + fibonacci (n - 1)
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| </source>
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| [[Image:Fib 4.png|thumb|300px|Call tree for <code>fibonacci 4</code>.]]
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| This function, again applied to any valid input, will generate a call tree which is non-linear. In the example on the right, the call tree generated by applying the <code>fibonacci</code> function to the input <code>4</code>.
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| This time, the anamorphism is the generation of the call tree isomorphic to the tree with [[leaf node]]s <code>0, 1, 1, 0, 1</code> and the catamorphism the [[summation]] of these leaf nodes.
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| ==See also==
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| * [[Apomorphism]]
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| * [[Paramorphism]]
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| * [[Morphism]]
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| ==References==
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| * {{cite web
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| |url=http://citeseer.ist.psu.edu/cachedpage/214108/1
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| |title=Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire
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| |author=Erik Meijer, Maarten Fokkinga, Ross Paterson
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| |pages=4, 5
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| |year=1991
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| }}
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| ==External links==
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| * [http://ulissesaraujo.wordpress.com/2009/04/09/hylomorphisms-in-haskell/ Hylomorphisms in Haskell]
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| * [http://ulissesaraujo.wordpress.com/2009/04/09/more-hylomorphisms-in-haskell/ More Hylomorphisms in Haskell]
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| [[Category:Articles with example Haskell code]]
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| [[Category:Category theory]]
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| [[Category:Recursion schemes]]
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