Probabilistic automaton - Revision history

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2014-05-27T23:12:10Z

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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.~~		Friends call him Royal. She is currently a cashier but quickly she'll be on her personal. Some time in the past I selected to reside in Arizona but I require to move for my family members. Climbing is what love performing.<br><br>Here is my homepage :: extended car warranty ([http://creativehomes.net/ActivityFeed/MyProfile/tabid/60/userId/69085/Default.aspx click the following web page])

	~~==Formal definition==~~

	~~A hylomorphism <math>h : A \rightarrow C</math> can~~ be ~~defined in terms of its separate anamorphic and catamorphic parts~~.

	~~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.~~

	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.~~

	~~Thus a hylomorphism~~
	~~:<math>~~
	~~h\,a = \begin{cases}~~
	~~c & \mbox{if } p\,a~~
	~~\\b \oplus ha' & \mbox{otherwise}~~
	~~\end{cases}~~
	~~</math>~~
	~~(where <math>(b, a') = ga</math>) may be defined (assuming appropriate definitions of <math>p</math> & <math>g</math>).~~

	~~===Notation===~~
	~~An abbreviated notation~~ for ~~the above hylomorphism is <math>h = [\![(c, \oplus),(g, p)]\!]</math>.~~

	~~==Hylomorphisms in practice==~~
	~~===Lists===~~
	[[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.

	~~One example of a commonly encountered hylomorphism is the canonical [[factorial]] function.~~

	~~<source lang="haskell">~~
	~~factorial :: Integer -> Integer~~
	~~factorial n~~
	~~\| n == 0 = 1~~
	~~\| n > 0 = n * factorial (n - 1)~~
	~~</source>~~

	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:~~

	~~factorial 5 = 5 * (factorial 4) = 120~~
	~~factorial 4 = 4 * (factorial 3) = 24~~
	~~factorial 3 = 3 * (factorial 2) = 6~~
	~~factorial 2 = 2 * (factorial 1) = 2~~
	~~factorial 1 = 1 * (factorial 0) = 1~~
	~~factorial 0 = 1~~

	~~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~~>.

	~~===Trees===~~

	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]].~~

	~~<source lang="haskell">~~
	~~fibonacci~~ :: ~~Integer -> Integer~~
	~~fibonacci n~~
	~~\| n == 0 = 0~~
	~~\| n == 1 = 1~~
	~~\| n > 1 = fibonacci (n - 2) + fibonacci~~ (~~n - 1)~~
	~~</source>~~

	~~[[Image:Fib 4.png\|thumb\|300px\|Call tree for <code>fibonacci 4</code>.]]~~

	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>.

	~~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.~~

	~~==See also==~~
	* [[Apomorphism]]
	* [[Paramorphism]]
	* [[Morphism]]

	~~==References==~~
	* {{cite web
	~~\|url=http://citeseer.ist.psu.edu/cachedpage/214108/1~~
	~~\|title=Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire~~
	~~\|author=Erik Meijer, Maarten Fokkinga, Ross Paterson~~
	~~\|pages=4, 5~~
	~~\|year=1991~~
	}}

	~~==External links==~~
	* [http://~~ulissesaraujo.wordpress~~.~~com~~/~~2009~~/04/09/~~hylomorphisms-in-haskell~~/ ~~Hylomorphisms in Haskell]~~
	* [http://~~ulissesaraujo~~.~~wordpress.com/2009/04/09/more-hylomorphisms-in-haskell/ More Hylomorphisms in Haskell]~~

	~~[[Category:Articles with example Haskell code]]~~
	~~[[Category:Category theory]]~~
	~~[[Category:Recursion schemes]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q176567

2013-03-17T02:35:43Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q176567

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	~~Hi there~~, ~~I am Sophia~~. ~~Office supervising~~ is ~~what she does for~~ a ~~residing~~. ~~For years he~~'~~s been living~~ in ~~Mississippi~~ and ~~he doesn~~'~~t strategy on altering it~~. The ~~favorite hobby~~ for ~~him~~ and ~~his children~~ is to ~~play lacross and he would by no means give~~ it up.<br><br>~~Feel free~~ to ~~surf~~ to ~~my weblog~~ :: ~~online psychic readings~~ [[http://~~ltreme~~.com/~~index~~.~~php?do=~~/~~profile~~-~~127790~~/~~info/ Recommended Internet page~~]]		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.

			==Formal definition==

			A hylomorphism <math>h : A \rightarrow C</math> can be defined in terms of its separate anamorphic and catamorphic parts.

			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.

			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.

			Thus a hylomorphism
			:<math>
			h\,a = \begin{cases}
			c & \mbox{if } p\,a
			\\b \oplus ha' & \mbox{otherwise}
			\end{cases}
			</math>
			(where <math>(b, a') = ga</math>) may be defined (assuming appropriate definitions of <math>p</math> & <math>g</math>).

			===Notation===
			An abbreviated notation for the above hylomorphism is <math>h = [\![(c, \oplus),(g, p)]\!]</math>.

			==Hylomorphisms in practice==
			===Lists===
			[[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.

			One example of a commonly encountered hylomorphism is the canonical [[factorial]] function.

			<source lang="haskell">
			factorial :: Integer -> Integer
			factorial n
			\| n == 0 = 1
			\| n > 0 = n * factorial (n - 1)
			</source>

			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:

			factorial 5 = 5 * (factorial 4) = 120
			factorial 4 = 4 * (factorial 3) = 24
			factorial 3 = 3 * (factorial 2) = 6
			factorial 2 = 2 * (factorial 1) = 2
			factorial 1 = 1 * (factorial 0) = 1
			factorial 0 = 1

			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>.

			===Trees===

			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]].

			<source lang="haskell">
			fibonacci :: Integer -> Integer
			fibonacci n
			\| n == 0 = 0
			\| n == 1 = 1
			\| n > 1 = fibonacci (n - 2) + fibonacci (n - 1)
			</source>

			[[Image:Fib 4.png\|thumb\|300px\|Call tree for <code>fibonacci 4</code>.]]

			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>.

			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.

			==See also==
			* [[Apomorphism]]
			* [[Paramorphism]]
			* [[Morphism]]

			==References==
			* {{cite web
			\|url=http://citeseer.ist.psu.edu/cachedpage/214108/1
			\|title=Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire
			\|author=Erik Meijer, Maarten Fokkinga, Ross Paterson
			\|pages=4, 5
			\|year=1991
			}}

			==External links==
			* [http://ulissesaraujo.wordpress.com/2009/04/09/hylomorphisms-in-haskell/ Hylomorphisms in Haskell]
			* [http://ulissesaraujo.wordpress.com/2009/04/09/more-hylomorphisms-in-haskell/ More Hylomorphisms in Haskell]

			[[Category:Articles with example Haskell code]]
			[[Category:Category theory]]
			[[Category:Recursion schemes]]

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