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A recursively-enumerable set of natural numbers (cf. [[Enumerable set|Enumerable set]]) whose complement is an [[Immune set|immune set]]. Simple sets are intermediate in the sense of so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853101.png" />-reducibility (cf. [[Recursive set theory|Recursive set theory]]) between solvable sets and creative sets. The latter are the largest among the enumerable sets in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853102.png" />-reducibility. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853103.png" /> be an arbitrary simple set, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853104.png" /> be an arbitrary creative set of natural numbers (e.g. the set of Gödel numbers of theorems of formal arithmetic); then there does not exist a [[General recursive function|general recursive function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853105.png" /> reducing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853106.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853107.png" />, i.e. such that
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A recursively-enumerable set of natural numbers (cf. [[Enumerable set]]) whose complement is an [[immune set]]. Simple sets are intermediate in the sense of [[Many-one reducibility|$m$-reducibility]] (cf. [[Recursive set theory]]) between [[solvable set]]s and [[creative set]]s. The latter are the largest among the enumerable sets in the sense of $m$-reducibility. Let $P$ be an arbitrary simple set, and let $K$ be an arbitrary creative set of natural numbers (e.g. the set of Gödel numbers of theorems of formal arithmetic); then there does not exist a [[general recursive function]] $f$ reducing $K$ to $P$, i.e. such that
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$$
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x \in K \Leftrightarrow f(x) \in P \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853108.png" /></td> </tr></table>
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Reducibility of $P$ to $K$ always takes place, but not one solvable set is reducible to $K$.
  
Reducibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s0853109.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s08531010.png" /> always takes place, but not one solvable set is reducible to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085310/s08531011.png" />.
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Uspenskii,  "Leçons sur les fonctions calculables" , Hermann  (1966)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algorithms and recursive functions" , Wolters-Noordhoff  (1970)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  H. Rogers jr.,  "Theory of recursive functions and effective computability" , McGraw-Hill  (1967)  pp. 164–165</TD></TR>
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</table>
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====Comments====
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A set is ''hypersimple'' if it is recursively enumerable and its complement is a [[hyperimmune set]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Uspenskii,  "Leçons sur les fonctions calculables" , Hermann  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algorithms and recursive functions" , Wolters-Noordhoff  (1970)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Rogers jr.,  "Theory of recursive functions and effective computability" , McGraw-Hill  (1967)  pp. 164–165</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> Nies, André. Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford University Press (2009). ISBN 978-0-19-923076-1. Zbl 1169.03034</TD></TR>
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</table>
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{{TEX|done}}
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[[Category:Computability and recursion theory]]

Latest revision as of 12:32, 17 January 2016

A recursively-enumerable set of natural numbers (cf. Enumerable set) whose complement is an immune set. Simple sets are intermediate in the sense of $m$-reducibility (cf. Recursive set theory) between solvable sets and creative sets. The latter are the largest among the enumerable sets in the sense of $m$-reducibility. Let $P$ be an arbitrary simple set, and let $K$ be an arbitrary creative set of natural numbers (e.g. the set of Gödel numbers of theorems of formal arithmetic); then there does not exist a general recursive function $f$ reducing $K$ to $P$, i.e. such that $$ x \in K \Leftrightarrow f(x) \in P \ . $$

Reducibility of $P$ to $K$ always takes place, but not one solvable set is reducible to $K$.

References

[1] V.A. Uspenskii, "Leçons sur les fonctions calculables" , Hermann (1966) (Translated from Russian)
[2] A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russian)
[3] H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165

Comments

A set is hypersimple if it is recursively enumerable and its complement is a hyperimmune set.

References

[a1] Nies, André. Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford University Press (2009). ISBN 978-0-19-923076-1. Zbl 1169.03034
How to Cite This Entry:
Simple set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple_set&oldid=18731
This article was adapted from an original article by S.N. Artemov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article