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en>Yobot: Tagging using AWB (10468)

2014-09-18T18:34:33Z

Tagging using AWB (10468)

← Older revision		Revision as of 20:34, 18 September 2014
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	~~In [[recursion theory]] a subset of~~ the ~~natural numbers is called a '''simple set''' if it is co-infinite and [[recursively enumerable]]~~, ~~but every infinite subset of its complement fails to be enumerated recursively.~~ ~~Simple sets are examples of recursively enumerable sets that are not~~ [~~[computable set\|recursive]]~~.		Hi there. Allow me begin by introducing the author, her [http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 psychic phone] title is Sophia Boon but she by no means truly favored that title. The preferred hobby for him and his kids is to perform lacross and free psychic ([http://203.250.78.160/zbxe/?document_srl=1792908 http://203.250.78.160/zbxe/?document_srl=1792908]) he'll be starting something else along with it. North Carolina is where we've been residing for many years and will never move. Office supervising is where her main earnings arrives from.<br><br>my blog post: cheap psychic readings ([http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- Visit medialab.zendesk.com])

	~~== Relation to Post's problem~~ ==
	~~Simple sets were devised by [[Emil Leon Post~~]~~] in the search for a non-Turing-complete recursively enumerable set. Whether such sets exist~~ is ~~known as [[Post's problem]]~~. ~~Post had to prove two things in order to obtain~~ his ~~result, one~~ is ~~that the simple set, say ''A'', does not Turing-reduce~~ to ~~the empty set,~~ and ~~that the ''K'', the [[halting problem]], does not Turing-reduce to ''A''.~~ ~~He succeeded in the first part~~ (~~which is obvious by definition), but for the other part, he managed only to prove a~~ [~~[many-one reduction]]~~.

	~~It was affirmed by Friedberg and Muchnik in the 1950s using a novel technique called the [[priority method]]~~. ~~They give a construction for a set that is simple (and thus non-recursive), but fails to compute the halting problem~~.~~<ref name=Nies35>Nies (2009) p.35<~~/~~ref>~~

	=~~= Formal definitions and some properties ==~~
	*A set <math>I \subseteq \mathbb{N}</~~math> is called '''immune''' iff <math>I</math> is infinite, but for every index <math>e<~~/~~math>, we have <math>W_e \text{ infinite} \implies W_e \not\subseteq I</math>~~. ~~Or equivalently: there is no infinite subset of <math>I</math> that is recursively enumerable~~.
	*A set <math>S \subseteq \mathbb{N}</math> is called ''' simple ''' iff it is recursively enumerable and its complement is immune.
	*A set <math>I \subseteq \mathbb{N}</~~math> is called '''effectively immune''' iff <math>I<~~/~~math> is infinite, but there exists a recursive function <math>f</math> such that for every index <math>e</math>, we have that <math> W_e \subseteq I \implies \#(W_e~~) ~~< f(e)</math>.~~
	*A set <math>S \subseteq \mathbb{N}</math> is called '~~''effectively simple''' if~~ it ~~is recursively enumerable and its complement is effectively immune~~. ~~Every effectively simple set,~~ is ~~simple~~ and ~~Turing-complete~~.
	*A set <math>I \subseteq \mathbb{N}</math> is called '''hyperimmune''' iff <math>I</math> is infinite, but <math>p_I</math> is ~~not computably dominated,~~ where <~~math~~>~~p_I~~<~~/math~~> ~~is the list of members of <math>I</math> in order.<ref name=Nies27>Nies~~ (~~2009) p.27<~~/~~ref>~~
	*A set <math>S \subseteq \mathbb{N}</~~math> is called '''hypersimple''' if it is simple and its complement is hyperimmune~~.~~<ref name=Nies37>Nies (2009) p~~.~~37<~~/~~ref>~~

	~~== Notes ==~~
	~~{{reflist}}~~

	~~== References ==~~
	* {{cite book \| first=Robert I. \| last=Soare \| title=Recursively enumerable sets and degrees. A study of computable functions and computably generated sets \| series=Perspectives in Mathematical Logic \| publisher=[[Springer-~~Verlag]] \| location=Berlin \| year=1987 \| isbn=3~~-~~540~~-~~15299~~-~~7 \| zbl=0667.03030 }}~~
	* {{cite book \| first=Piergiorgio \| last=Odifreddi \| authorlink=Piergiorgio Odifreddi \| title=Classical recursion theory. The theory of functions and sets of natural numbers \| publisher=North Holland \| year=1988 \| zbl=0661.03029 \| series=Studies in Logic and the Foundations of Mathematics \| volume=125 \| location=Amsterdam \| isbn=0-~~444~~-~~87295~~-~~7 }}~~
	* {{cite book \| last=Nies \| first=André \| title=Computability and randomness \| series=Oxford Logic Guides \| volume=51 \| location=Oxford \| publisher=Oxford University Press \| year=2009 \| isbn=978-0-19-~~923076~~-~~1 \| zbl=1169~~.~~03034 }}~~

	~~[[Category:Computability theory]~~]

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q453471

2013-03-02T10:54:52Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q453471

← Older revision		Revision as of 12:54, 2 March 2013
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	~~The author~~ is known by the name ~~of Figures Wunder~~. ~~North Dakota~~ is ~~our birth place~~. ~~Body building~~ is ~~what my family and~~ I ~~enjoy~~. ~~Hiring~~ is ~~her day job now but she~~'~~s always wanted her personal business~~.<br><br>~~Take~~ a ~~look at my blog [http:~~//~~immbooks~~.~~com~~/~~blogs~~/~~post~~/~~44599 http:~~//~~immbooks~~.~~com~~/]		In [[recursion theory]] a subset of the natural numbers is called a '''simple set''' if it is co-infinite and [[recursively enumerable]], but every infinite subset of its complement fails to be enumerated recursively. Simple sets are examples of recursively enumerable sets that are not [[computable set\|recursive]].

			== Relation to Post's problem ==
			Simple sets were devised by [[Emil Leon Post]] in the search for a non-Turing-complete recursively enumerable set. Whether such sets exist is known as [[Post's problem]]. Post had to prove two things in order to obtain his result, one is that the simple set, say ''A'', does not Turing-reduce to the empty set, and that the ''K'', the [[halting problem]], does not Turing-reduce to ''A''. He succeeded in the first part (which is obvious by definition), but for the other part, he managed only to prove a [[many-one reduction]].

			It was affirmed by Friedberg and Muchnik in the 1950s using a novel technique called the [[priority method]]. They give a construction for a set that is simple (and thus non-recursive), but fails to compute the halting problem.<ref name=Nies35>Nies (2009) p.35</ref>

			== Formal definitions and some properties ==
			*A set <math>I \subseteq \mathbb{N}</math> is called '''immune''' iff <math>I</math> is infinite, but for every index <math>e</math>, we have <math>W_e \text{ infinite} \implies W_e \not\subseteq I</math>. Or equivalently: there is no infinite subset of <math>I</math> that is recursively enumerable.
			*A set <math>S \subseteq \mathbb{N}</math> is called ''' simple ''' iff it is recursively enumerable and its complement is immune.
			*A set <math>I \subseteq \mathbb{N}</math> is called '''effectively immune''' iff <math>I</math> is infinite, but there exists a recursive function <math>f</math> such that for every index <math>e</math>, we have that <math> W_e \subseteq I \implies \#(W_e) < f(e)</math>.
			*A set <math>S \subseteq \mathbb{N}</math> is called '''effectively simple''' if it is recursively enumerable and its complement is effectively immune. Every effectively simple set, is simple and Turing-complete.
			*A set <math>I \subseteq \mathbb{N}</math> is called '''hyperimmune''' iff <math>I</math> is infinite, but <math>p_I</math> is not computably dominated, where <math>p_I</math> is the list of members of <math>I</math> in order.<ref name=Nies27>Nies (2009) p.27</ref>
			*A set <math>S \subseteq \mathbb{N}</math> is called '''hypersimple''' if it is simple and its complement is hyperimmune.<ref name=Nies37>Nies (2009) p.37</ref>

			== Notes ==
			{{reflist}}

			== References ==
			* {{cite book \| first=Robert I. \| last=Soare \| title=Recursively enumerable sets and degrees. A study of computable functions and computably generated sets \| series=Perspectives in Mathematical Logic \| publisher=[[Springer-Verlag]] \| location=Berlin \| year=1987 \| isbn=3-540-15299-7 \| zbl=0667.03030 }}
			* {{cite book \| first=Piergiorgio \| last=Odifreddi \| authorlink=Piergiorgio Odifreddi \| title=Classical recursion theory. The theory of functions and sets of natural numbers \| publisher=North Holland \| year=1988 \| zbl=0661.03029 \| series=Studies in Logic and the Foundations of Mathematics \| volume=125 \| location=Amsterdam \| isbn=0-444-87295-7 }}
			* {{cite book \| last=Nies \| first=André \| title=Computability and randomness \| series=Oxford Logic Guides \| volume=51 \| location=Oxford \| publisher=Oxford University Press \| year=2009 \| isbn=978-0-19-923076-1 \| zbl=1169.03034 }}

			[[Category:Computability theory]]

en>Maxim: /* See also */ +cat

2012-06-03T01:57:17Z