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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]].
 
== 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]]

Revision as of 11:54, 2 March 2013

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 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.[1]

Formal definitions and some properties

  • A set I⊆ℕ is called immune iff I is infinite, but for every index e, we have We infinite⟹We⊈I. Or equivalently: there is no infinite subset of I that is recursively enumerable.
  • A set S⊆ℕ is called simple iff it is recursively enumerable and its complement is immune.
  • A set I⊆ℕ is called effectively immune iff I is infinite, but there exists a recursive function f such that for every index e, we have that We⊆I⟹#(We)<f(e).
  • A set S⊆ℕ 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 I⊆ℕ is called hyperimmune iff I is infinite, but pI is not computably dominated, where pI is the list of members of I in order.[2]
  • A set S⊆ℕ is called hypersimple if it is simple and its complement is hyperimmune.[3]

Notes

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References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  1. ↑ Nies (2009) p.35
  2. ↑ Nies (2009) p.27
  3. ↑ Nies (2009) p.37
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