Simple set

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 $I\subseteq {\mathbb {N}}$ is called immune if $I$ is infinite, but for every index $e$ , we have $W_{e}{\text{ infinite}}\implies W_{e}\not \subseteq I$ . Or equivalently: there is no infinite subset of $I$ that is recursively enumerable.
A set $S\subseteq {\mathbb {N}}$ is called simple if it is recursively enumerable and its complement is immune.
A set $I\subseteq {\mathbb {N}}$ is called effectively immune if $I$ is infinite, but there exists a recursive function $f$ such that for every index $e$ , we have that $W_{e}\subseteq I\implies \#(W_{e})<f(e)$ .
A set $S\subseteq {\mathbb {N}}$ 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 $I\subseteq {\mathbb {N}}$ is called hyperimmune if $I$ is infinite, but $p_{I}$ is not computably dominated, where $p_{I}$ is the list of members of $I$ in order.^[2]
A set $S\subseteq {\mathbb {N}}$ is called hypersimple if it is simple and its complement is hyperimmune.^[3]

Notes

↑ Nies (2009) p.35
↑ Nies (2009) p.27
↑ Nies (2009) p.37

References

Soare, Robert I. (1987). Recursively enumerable sets and degrees. A study of computable functions and computably generated sets. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. ISBN 3-540-15299-7. Zbl 0667.03030.
Odifreddi, Piergiorgio (1988). Classical recursion theory. The theory of functions and sets of natural numbers. Studies in Logic and the Foundations of Mathematics. 125. Amsterdam: North Holland. ISBN 0-444-87295-7. Zbl 0661.03029.
Nies, André (2009). Computability and randomness. Oxford Logic Guides. 51. Oxford: Oxford University Press. ISBN 978-0-19-923076-1. Zbl 1169.03034.

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[Nies35-1] Nies (2009) p.35

[Nies27-2] Nies (2009) p.27

[Nies37-3] Nies (2009) p.37