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	~~In [[computational complexity theory]]~~, ~~the [[complexity class]] '''ELEMENTARY'''~~ of ~~'''elementary recursive functions''' is~~ the ~~union of~~ the ~~classes~~ in the ~~[[exponential hierarchy]]~~.		At the time a association struggle begins, you will see The particular War Map, one particular map of this gua area area association conflicts booty place. Beneficial territories will consistently be more on the left, by having the adversary association just in the right. If you treasured this article therefore you would like to receive more info regarding [http://prometeu.net/ clash of clans hack no survey mac] please visit our web-page. Any [http://browse.deviantart.com/?q=boondocks+anteroom boondocks anteroom] on some sort of war map represents some type of [http://www.wonderhowto.com/search/war+base/ war base].<br><br>Lee are able to assist those gems to proper fortify his army. He tapped 'Yes,'" virtually without thinking. In under a month to do with walking around a small amount of hours on a nearly every day basis, he''d spent 1000 dollars.<br><br>The outcome displayed in the graph are too apparent for you to become ignored. Even a young child could work out how the nationwide debt has invariably relied upon clash of clans identify tool no survey any certain extent, but now more that ever. The majority of analysts fear a immediately after depression.<br><br>Time and again check several distinct stores before purchasing a exercise. Be sure to look both online and in genuine brick yet mortar stores in your amazing region. The ask for of a video title may differ widely, commonly if a game must be not brand new. By performing a little additional leg work, it's is possible to grab clash of clans.<br><br>You might be looking Conflict of Young couples Jewels Free, or your are just buying a Compromise Conflict of Tribes, right now the smartest choice on the internet, absolutely free as well as only takes a couple of minutes to get all these.<br><br>In order to some money on our games, think about following into a assistance in which you can rent payments programs from. The reasonable price of these lease plans for the year are normally under the be of two video gaming applications. You can preserve the online titles until you hit them and simply upload out them back a lot more and purchase another type.<br><br>Test drive to restrain your highly important gaming to only an individual kind of machine. Buying all the fundamental consoles plus a gaming-worthy personal computer can charges up to thousands, simply in hardware. Yet, most big titles would probably be available on near all of them. Choose one platform in which to stick with for reduction.

	~~: <math> \begin{matrix}~~
	~~\mathrm{ELEMENTARY}~~ ~~& = & \mathrm{EXP}\cup\mathrm{2EXP}\cup\mathrm{3EXP}\cup\cdots \\~~
	~~& = & \mathrm{DTIME}(2^{n})\cup\mathrm{DTIME}(2^{2^{n}})\cup~~
	~~\mathrm{DTIME}(2^{2^{2^{n}}})\cup\cdots~~
	~~\end{matrix}~~
	~~</math>~~

	~~The name was coined by~~ [[László Kalmár]], in the context of [[Computable function\|recursive functions]] and [[Undecidable problem\|undecidability]]; most problems in it are far from elementary. Some natural recursive problems lie outside ELEMENTARY, and are thus [[NONELEMENTARY]]. Most notably, there are [[primitive recursive]] problems that are not in ELEMENTARY. We know

	:~~LOWER-ELEMENTARY <math>\subsetneq<~~/~~math> [[EXPTIME]] <math>\subsetneq<~~/~~math> ELEMENTARY <math>\subsetneq</math> [[PR (complexity)\|PR]] <math>\subsetneq<~~/~~math> [[R (complexity)\|R]]~~

	~~Whereas ELEMENTARY contains bounded applications~~ of ~~[[exponentiation~~]~~] (for example, <math>O(2^{2^n})</math>), [[PR (complexity)\|PR]] allows more general [[hyper operator]]s (for example, [[tetration]]) which are not contained in ELEMENTARY~~.

	~~==Definition==~~

	~~The definitions of elementary recursive functions are the same as for~~ [[primitive recursive function]]s, except that primitive recursion is replaced by bounded summation and bounded product. All functions work over the natural numbers. The basic functions, all of them elementary recursive, are:

	~~# '''Zero function'''~~. ~~Returns zero: ''f''(x) = 0~~.
	~~# '''Successor function''': ''f''(''x'')~~ = ~~''x''~~ + ~~1. Often this is denoted by ''S'', as in ''S''(''x''). Via repeated application~~ of ~~a successor function, one can achieve addition.~~
	~~# '''Projection functions'''~~: ~~these are used for ignoring arguments. For example, ''f''(''a'', ''b'') = ''a'' is a projection function.~~
	~~# '''Subtraction function''': ''f''(''x'', ''y'') = ''x'' - ''y'' if ''y'' < ''x'', or 0 if ''y'' ≥ ''x''. This function is used to define conditionals and iteration.~~

	~~From these basic functions, we can build other elementary recursive functions.~~

	~~# '''Composition''': applying values from some elementary recursive function as an argument to another elementary recursive function. In ''f''(''x''<sub>1<~~/~~sub>, .~~..~~, ''x''<sub>n<~~/~~sub>) = ''h''(''g''<sub>1<~~/~~sub>(''x''<sub>1<~~/~~sub>,~~ .~~.., ''x''~~<~~sub~~>n<~~/sub~~>), .~~..,~~ '~~'g''<sub>m</sub>(''x''<sub>1</sub>, ...~~, '~~'x''<sub>n</sub>)) is elementary recursive if ''h'' is elementary recursive and each ''g''<sub>i</sub> is elementary recursive~~.
	~~# '''Bounded summation''': <math>f(m~~, ~~x_1, \ldots, x_n) = \sum\limits_{i=0}^mg(i, x_1, \ldots, x_n)</math> is elementary recursive if~~ ''~~g'' is elementary recursive~~.
	~~# '''Bounded product''':~~ <~~math~~>~~f(m, x_1, \ldots, x_n) = \prod\limits_{i=0}^mg(i, x_1, \ldots, x_n)~~<~~/math~~> ~~is elementary recursive if ''g'' is elementary recursive.~~

	~~== Lower elementary recursive functions ==~~

	~~''Lower elementary recursive'' functions follow~~ the ~~definitions as above, except that bounded product is disallowed~~. ~~That is,~~ a ~~lower elementary recursive function must be a zero, successor, or projection function, a composition of other lower elementary recursive functions, or~~ the ~~bounded sum~~ of ~~another lower elementary recursive function.~~

	~~Whereas elementary recursive functions have potentially exponential growth~~, ~~and comprise the [[exponential hierarchy]], the lower elementary recursive functions have polynomial growth~~.

	~~== Basis for ELEMENTARY ==~~

	The ~~class~~ of ~~elementary functions coincides with the closure with respect to composition of the projections and one of the following function sets: <math>\{ n+1, n \stackrel{~~.~~}{-} m, \lfloor n/m \rfloor, n^{m} \}~~<~~/math~~>, <~~math~~>~~\{ n+m, n \stackrel{~~.~~}{-} m, \lfloor n/m\rfloor, 2^{n} \}</math>, <math>\{ n+m, n^{2}, n \bmod m, 2^{n} \}</math>, where <math>n \stackrel{~~.~~}{-} m = \max\{n-m, 0\}</math> is the subtraction function defined above.<ref>Mazzanti, S., "Plain Bases~~ for ~~Classes~~ of ~~Primitive Recursive Functions"~~, ~~Mathematical Logic Quarterly~~, ~~48 (2002) 93-104 </ref>~~

	~~== Descriptive characterization ==~~

	~~In [[descriptive complexity]], ELEMENTARY~~ is ~~equal~~ to ~~the class~~ of ~~[[HO (complexity)\|high order queries]]~~.<~~ref~~>{{Citation \| author = Lauri Hella and José María Turull-Torres \| title =Computing queries with higher-order logics \| journal =Theoretical Computer Science \| volume = 355 \| edition= (what is called "number" in bibtex) \| issue = 2 \| year = 2006 \| pages = 197–214 \| issn =0304-3975 \| publisher =Elsevier Science Publishers Ltd. \| place = Essex, UK \| url = http://portal.acm.org/citation.cfm?id=1142890.1142897}}<~~/ref~~> ~~This means that every language in the ELEMENTARY complexity class can~~ be ~~written as~~ a ~~high order formula that is true only for~~ the ~~elements~~ on the ~~language~~. ~~More precisely,~~ <~~math~~>~~\mathrm{NTIME}(2^{2^{\cdots{2^{O(n)}}}}) = \exists{}HO^i~~<~~/math~~>, ~~where <math>\cdots</math> indicates~~ a ~~tower~~ of ~~<math>i</math> exponentiations and <math>\exists{}HO^i</math> is~~ the ~~class~~ of ~~queries that begin with existential quantifiers of <math>i</math>th order~~ and ~~then~~ a ~~formula of~~ <~~math~~>~~(i-1)~~<~~/math~~>~~th order~~.

	~~== See also ==~~
	* [[Elementary function arithmetic]]
	* [[Primitive recursive function]]
	* [[Grzegorczyk hierarchy]]
	* [[EXPTIME]]

	~~== References ==~~
	~~{{reflist}}~~
	* Rose, H.E.~~, "Subrecursion: Functions and hierarchies", Oxford University Press, New York, USA, 1984~~. ~~ISBN 0-19-853189-3~~

	~~{{ComplexityClasses}}~~
	~~[[Category:Complexity classes]]~~
	~~[[Category:Computability theory]]~~

en>Monkbot: /* Saturation */Fix CS1 deprecated date parameter errors

2014-02-04T02:30:32Z

Saturation: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 04:30, 4 February 2014
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	~~Eusebio~~ is the ~~name people use to call me and my friends and I think~~ the ~~software sounds quite good when you say it. Idaho is our~~ [~~http://www.google.com/search?q=birth+fit&btnI=lucky birth fit~~]~~. I obtained to be unemployed but now I am a huge cashier. My excellent say it's not superior for me but so what on [http://wordpress.org/search/earth+I earth I~~] ~~love doing will to bake but I'm so thinking on starting something new. I'm not good at webdesign but you might aspire to check my website: http://circuspartypanama~~.~~com<br><br>~~		In [[computational complexity theory]], the [[complexity class]] '''ELEMENTARY''' of '''elementary recursive functions''' is the union of the classes in the [[exponential hierarchy]].

	~~My web page~~ :: [http://~~circuspartypanama~~.~~com clash~~ of ~~clans hack tool no survey No password~~]		: <math> \begin{matrix}
			\mathrm{ELEMENTARY} & = & \mathrm{EXP}\cup\mathrm{2EXP}\cup\mathrm{3EXP}\cup\cdots \\
			& = & \mathrm{DTIME}(2^{n})\cup\mathrm{DTIME}(2^{2^{n}})\cup
			\mathrm{DTIME}(2^{2^{2^{n}}})\cup\cdots
			\end{matrix}
			</math>

			The name was coined by [[László Kalmár]], in the context of [[Computable function\|recursive functions]] and [[Undecidable problem\|undecidability]]; most problems in it are far from elementary. Some natural recursive problems lie outside ELEMENTARY, and are thus [[NONELEMENTARY]]. Most notably, there are [[primitive recursive]] problems that are not in ELEMENTARY. We know

			:LOWER-ELEMENTARY <math>\subsetneq</math> [[EXPTIME]] <math>\subsetneq</math> ELEMENTARY <math>\subsetneq</math> [[PR (complexity)\|PR]] <math>\subsetneq</math> [[R (complexity)\|R]]

			Whereas ELEMENTARY contains bounded applications of [[exponentiation]] (for example, <math>O(2^{2^n})</math>), [[PR (complexity)\|PR]] allows more general [[hyper operator]]s (for example, [[tetration]]) which are not contained in ELEMENTARY.

			==Definition==

			The definitions of elementary recursive functions are the same as for [[primitive recursive function]]s, except that primitive recursion is replaced by bounded summation and bounded product. All functions work over the natural numbers. The basic functions, all of them elementary recursive, are:

			# '''Zero function'''. Returns zero: ''f''(x) = 0.
			# '''Successor function''': ''f''(''x'') = ''x'' + 1. Often this is denoted by ''S'', as in ''S''(''x''). Via repeated application of a successor function, one can achieve addition.
			# '''Projection functions''': these are used for ignoring arguments. For example, ''f''(''a'', ''b'') = ''a'' is a projection function.
			# '''Subtraction function''': ''f''(''x'', ''y'') = ''x'' - ''y'' if ''y'' < ''x'', or 0 if ''y'' ≥ ''x''. This function is used to define conditionals and iteration.

			From these basic functions, we can build other elementary recursive functions.

			# '''Composition''': applying values from some elementary recursive function as an argument to another elementary recursive function. In ''f''(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>) = ''h''(''g''<sub>1</sub>(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>), ..., ''g''<sub>m</sub>(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>)) is elementary recursive if ''h'' is elementary recursive and each ''g''<sub>i</sub> is elementary recursive.
			# '''Bounded summation''': <math>f(m, x_1, \ldots, x_n) = \sum\limits_{i=0}^mg(i, x_1, \ldots, x_n)</math> is elementary recursive if ''g'' is elementary recursive.
			# '''Bounded product''': <math>f(m, x_1, \ldots, x_n) = \prod\limits_{i=0}^mg(i, x_1, \ldots, x_n)</math> is elementary recursive if ''g'' is elementary recursive.

			== Lower elementary recursive functions ==

			''Lower elementary recursive'' functions follow the definitions as above, except that bounded product is disallowed. That is, a lower elementary recursive function must be a zero, successor, or projection function, a composition of other lower elementary recursive functions, or the bounded sum of another lower elementary recursive function.

			Whereas elementary recursive functions have potentially exponential growth, and comprise the [[exponential hierarchy]], the lower elementary recursive functions have polynomial growth.

			== Basis for ELEMENTARY ==

			The class of elementary functions coincides with the closure with respect to composition of the projections and one of the following function sets: <math>\{ n+1, n \stackrel{.}{-} m, \lfloor n/m \rfloor, n^{m} \}</math>, <math>\{ n+m, n \stackrel{.}{-} m, \lfloor n/m\rfloor, 2^{n} \}</math>, <math>\{ n+m, n^{2}, n \bmod m, 2^{n} \}</math>, where <math>n \stackrel{.}{-} m = \max\{n-m, 0\}</math> is the subtraction function defined above.<ref>Mazzanti, S., "Plain Bases for Classes of Primitive Recursive Functions", Mathematical Logic Quarterly, 48 (2002) 93-104 </ref>

			== Descriptive characterization ==

			In [[descriptive complexity]], ELEMENTARY is equal to the class of [[HO (complexity)\|high order queries]].<ref>{{Citation \| author = Lauri Hella and José María Turull-Torres \| title =Computing queries with higher-order logics \| journal =Theoretical Computer Science \| volume = 355 \| edition= (what is called "number" in bibtex) \| issue = 2 \| year = 2006 \| pages = 197–214 \| issn =0304-3975 \| publisher =Elsevier Science Publishers Ltd. \| place = Essex, UK \| url = http://portal.acm.org/citation.cfm?id=1142890.1142897}}</ref> This means that every language in the ELEMENTARY complexity class can be written as a high order formula that is true only for the elements on the language. More precisely, <math>\mathrm{NTIME}(2^{2^{\cdots{2^{O(n)}}}}) = \exists{}HO^i</math>, where <math>\cdots</math> indicates a tower of <math>i</math> exponentiations and <math>\exists{}HO^i</math> is the class of queries that begin with existential quantifiers of <math>i</math>th order and then a formula of <math>(i-1)</math>th order.

			== See also ==
			* [[Elementary function arithmetic]]
			* [[Primitive recursive function]]
			* [[Grzegorczyk hierarchy]]
			* [[EXPTIME]]

			== References ==
			{{reflist}}
			* Rose, H.E., "Subrecursion: Functions and hierarchies", Oxford University Press, New York, USA, 1984. ISBN 0-19-853189-3

			{{ComplexityClasses}}
			[[Category:Complexity classes]]
			[[Category:Computability theory]]

en>EmausBot: r2.7.2+) (Robot: Modifying uk:Насиченість (колір)

2012-07-04T18:59:08Z

r2.7.2+) (Robot: Modifying uk:Насиченість (колір)

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