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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]].
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| == Relation to Post's problem ==
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| 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]].
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| 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>
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| == 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.
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| *A set <math>S \subseteq \mathbb{N}</math> is called ''' simple ''' iff it is recursively enumerable and its complement is immune.
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| *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>.
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| *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.
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| *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>
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| *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>
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{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 }}
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| * {{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 }}
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| * {{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 }}
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| [[Category:Computability theory]]
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