Thomson problem - Revision history

en>SporkBot: Replace template per TFD outcome; no change in content

2014-10-20T22:24:56Z

Replace template per TFD outcome; no change in content

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en>Tamfang: /* Configurations of smallest known energy */

2014-03-01T20:58:33Z

Configurations of smallest known energy

← Older revision		Revision as of 22:58, 1 March 2014
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	~~In [[computability theory]]~~ and [[computational complexity theory]], '''RE''' ([[Recursively enumerable set\|recursively enumerable]]) is the [[complexity class\|class]] of [[decision problem]]s for which a 'yes' answer can be verified by a [[Turing machine]] in a finite amount of time.~~<ref>{{CZoo\|Class RE\|R#re}}</ref> Informally, it means that if the answer is 'yes', then there is some procedure which takes finite time to determine this. On the other hand, if the answer~~ is ~~'no', the machine might never halt~~. ~~Equivalently, '''RE''' is the class~~ of ~~decision problems for which a Turing machine can list all~~ the ~~'yes' instances, one by one (this~~ is ~~what~~ '~~enumerable' means).~~		Hello and welcome. My title is Figures Wunder. One of the issues she loves most is to study comics and she'll be beginning some thing else along with it. Years ago we moved to North Dakota and I adore every day residing here. Bookkeeping is what I do.<br><br>Stop by my blog; [http://torontocartridge.com/uncategorized/the-ideal-way-to-battle-a-yeast-infection/ std home test]

	~~Similarly, '''co-RE''' is the set of all languages that are complements of a language in '''RE'''. In a sense, '''co-RE''' contains languages of which membership can~~ be ~~disproved in a finite amount of time, but proving membership might take forever~~.

	~~Each member of '''RE''' is a [[recursively enumerable set]]~~ and ~~therefore a [[Diophantine set]]~~.

	~~==Relations to other classes==~~
	~~The set of [[recursive language]]s ('''[[R (complexity)\|R]]''')~~ is ~~a subset of both '''RE''' and '''co-RE'''~~.<~~ref~~>~~{{CZoo\|Class co-RE\|C#core}}~~<~~/ref~~> ~~In fact, it is the intersection of those two classes, because we can decide any problem for which there exists a recogniser and also a co-recogniser~~ by ~~simply interleaving them until one obtains a result. Therefore~~:
	~~:<math>\mbox{R} = \mbox{RE}\cap\mbox{co-RE}.<~~/~~math>~~

	~~==RE-complete==~~
	~~'''RE-complete''' is the set of decision problems that are complete for '''RE'''. In a sense, these are the "hardest" recursively enumerable problems~~. ~~All such problems are non[[recursive language\|recursive]]. Generally, no constraint is placed on the reductions used except that they must be [[many-one reduction]]s.~~

	~~Examples of RE-complete problems:~~
	~~#[[Halting problem]]: Whether a program given a finite input finishes running or will run forever.~~
	#By [[Rice's Theorem]], deciding membership in any nontrivial subset of the set of [[Computable function\|recursive functions]] is '''RE'''-hard. It will be complete whenever the set is recursively enumerable.
	~~#{{harvs\|first=John\|last=Myhill\|authorlink=John Myhill\|year=1955\|txt}}<ref>{{citation~~
	~~\| last = Myhill \| first = John \| authorlink = John Myhill~~
	~~\| doi = 10.1002~~/~~malq.19550010205~~
	~~\| journal = Zeitschrift für Mathematische Logik und Grundlagen der Mathematik~~
	~~\| mr = 0071379~~
	~~\| pages = 97–108~~
	~~\| title = Creative sets~~
	~~\| volume = 1~~
	~~\| year = 1955}}.<~~/~~ref> has proven that all [[creative set]]s are '''RE'''-complete.~~
	#The uniform [[word problem (mathematics)\|word problem]] for [[group (mathematics)\|groups]] or [[semigroup]]s. [Indeed, the [[word problem for groups\|word problem for some individual groups]] is '''RE'''-complete.]
	~~#Deciding membership in a general [[unrestricted grammar\|unrestricted]] [[formal grammar]]. [Again, certain individual grammars have '''RE'''-complete membership problem.]~~
	~~#The [[validity]] problem for [[first-order logic]].~~
	~~#[[Post correspondence problem]]: Given a finite set of strings, determine if there is a string that can be factored into a composition of~~ the ~~strings (allowing repeats) in two different ways.~~
	~~#Determining if a [[Diophantine equation]] has any integer solutions.~~

	~~==co~~-RE-~~complete==~~
	~~'''co~~-RE-~~complete''' is the set of decision problems that are complete for '''co~~-~~RE'''. In~~ a ~~sense, these are the complements of the hardest recursively enumerable problems.~~

	~~Examples of co~~-RE-~~complete problems:~~
	~~#The [[Domino Problem]] for [[Wang tile]]s.~~
	~~#The [[satisfiability]] problem for [[first-order logic]]~~

	~~==See also==~~
	~~[[List of undecidable problems]]~~

	~~==References==~~
	~~<references~~/>

	~~{{ComplexityClasses}}~~

	~~[[Category:Complexity classes]~~]

en>SFK2: Reverted 2 edits by 66.41.36.136 (talk) identified as vandalism to last revision by ClueBot NG. (TW)

2012-10-14T23:38:06Z

Reverted 2 edits by 66.41.36.136 (talk) identified as vandalism to last revision by ClueBot NG. (TW)

← Older revision		Revision as of 01:38, 15 October 2012
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	~~The writer~~ is ~~known by~~ the ~~name~~ of ~~Numbers Wunder. One~~ of ~~the extremely best issues [http://www~~.~~shipzula.com~~/~~blogs/post/16377 home std test kit] in~~ the ~~world for him~~ is to ~~collect badges but he~~ is ~~having difficulties to find time~~ for it. ~~Bookkeeping~~ is ~~home std test kit her working day occupation now~~. [~~http://Www~~.~~marieclaire~~.~~com~~/~~health~~-~~fitness/weight~~-~~loss~~/at-~~home~~-~~sexually~~-~~transmitted~~-~~disease-testing Puerto Rico~~] ~~is where he~~'s ~~always been residing but she requirements to transfer simply because~~ of ~~her family~~.<br>~~<br>Here~~ ~~std testing at home is my~~ ~~at home std~~ [~~http://ironptstudio~~.~~com/rev/239375 over~~ the ~~counter std test~~] ~~testing web page~~: ~~home std test kit~~ - [~~http://xrambo~~.~~com~~/~~user/RKDBG check this link right here now~~] -		In [[computability theory]] and [[computational complexity theory]], '''RE''' ([[Recursively enumerable set\|recursively enumerable]]) is the [[complexity class\|class]] of [[decision problem]]s for which a 'yes' answer can be verified by a [[Turing machine]] in a finite amount of time.<ref>{{CZoo\|Class RE\|R#re}}</ref> Informally, it means that if the answer is 'yes', then there is some procedure which takes finite time to determine this. On the other hand, if the answer is 'no', the machine might never halt. Equivalently, '''RE''' is the class of decision problems for which a Turing machine can list all the 'yes' instances, one by one (this is what 'enumerable' means).

			Similarly, '''co-RE''' is the set of all languages that are complements of a language in '''RE'''. In a sense, '''co-RE''' contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever.

			Each member of '''RE''' is a [[recursively enumerable set]] and therefore a [[Diophantine set]].

			==Relations to other classes==
			The set of [[recursive language]]s ('''[[R (complexity)\|R]]''') is a subset of both '''RE''' and '''co-RE'''.<ref>{{CZoo\|Class co-RE\|C#core}}</ref> In fact, it is the intersection of those two classes, because we can decide any problem for which there exists a recogniser and also a co-recogniser by simply interleaving them until one obtains a result. Therefore:
			:<math>\mbox{R} = \mbox{RE}\cap\mbox{co-RE}.</math>

			==RE-complete==
			'''RE-complete''' is the set of decision problems that are complete for '''RE'''. In a sense, these are the "hardest" recursively enumerable problems. All such problems are non[[recursive language\|recursive]]. Generally, no constraint is placed on the reductions used except that they must be [[many-one reduction]]s.

			Examples of RE-complete problems:
			#[[Halting problem]]: Whether a program given a finite input finishes running or will run forever.
			#By [[Rice's Theorem]], deciding membership in any nontrivial subset of the set of [[Computable function\|recursive functions]] is '''RE'''-hard. It will be complete whenever the set is recursively enumerable.
			#{{harvs\|first=John\|last=Myhill\|authorlink=John Myhill\|year=1955\|txt}}<ref>{{citation
			\| last = Myhill \| first = John \| authorlink = John Myhill
			\| doi = 10.1002/malq.19550010205
			\| journal = Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
			\| mr = 0071379
			\| pages = 97–108
			\| title = Creative sets
			\| volume = 1
			\| year = 1955}}.</ref> has proven that all [[creative set]]s are '''RE'''-complete.
			#The uniform [[word problem (mathematics)\|word problem]] for [[group (mathematics)\|groups]] or [[semigroup]]s. [Indeed, the [[word problem for groups\|word problem for some individual groups]] is '''RE'''-complete.]
			#Deciding membership in a general [[unrestricted grammar\|unrestricted]] [[formal grammar]]. [Again, certain individual grammars have '''RE'''-complete membership problem.]
			#The [[validity]] problem for [[first-order logic]].
			#[[Post correspondence problem]]: Given a finite set of strings, determine if there is a string that can be factored into a composition of the strings (allowing repeats) in two different ways.
			#Determining if a [[Diophantine equation]] has any integer solutions.

			==co-RE-complete==
			'''co-RE-complete''' is the set of decision problems that are complete for '''co-RE'''. In a sense, these are the complements of the hardest recursively enumerable problems.

			Examples of co-RE-complete problems:
			#The [[Domino Problem]] for [[Wang tile]]s.
			#The [[satisfiability]] problem for [[first-order logic]]

			==See also==
			[[List of undecidable problems]]

			==References==
			<references/>

			{{ComplexityClasses}}

			[[Category:Complexity classes]]

en>Tamfang: /* Configurations of smallest known energy */ typo (997 for 977)

2012-05-30T18:27:00Z

Configurations of smallest known energy: typo (997 for 977)

New page

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