Simple set
In computability theory, a subset of the natural numbers is called simple if it is computably enumerable (c.e.) and co-infinite (i.e. its complement is infinite), but every infinite subset of its complement is not c.e.. Simple sets are examples of c.e. sets that are not computable.
Relation to Post's problem
Simple sets were devised by Emil Leon Post in the search for a non-Turing-complete c.e. set. Whether such sets exist is known as Post's problem. Post had to prove two things in order to obtain his result: that the simple set A is not computable, 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.
Post's idea was validated 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-computable), but fails to compute the halting problem.[1]
Formal definitions and some properties
In what follows, [math]\displaystyle{ W_e }[/math] denotes a standard uniformly c.e. listing of all the c.e. sets.
- A set [math]\displaystyle{ I \subseteq \mathbb{N} }[/math] is called immune if [math]\displaystyle{ I }[/math] is infinite, but for every index [math]\displaystyle{ e }[/math], we have [math]\displaystyle{ W_e \text{ infinite} \implies W_e \not\subseteq I }[/math]. Or equivalently: there is no infinite subset of [math]\displaystyle{ I }[/math] that is c.e..
- A set [math]\displaystyle{ S \subseteq \mathbb{N} }[/math] is called simple if it is c.e. and its complement is immune.
- A set [math]\displaystyle{ I \subseteq \mathbb{N} }[/math] is called effectively immune if [math]\displaystyle{ I }[/math] is infinite, but there exists a recursive function [math]\displaystyle{ f }[/math] such that for every index [math]\displaystyle{ e }[/math], we have that [math]\displaystyle{ W_e \subseteq I \implies \#(W_e) \lt f(e) }[/math].
- A set [math]\displaystyle{ S \subseteq \mathbb{N} }[/math] is called effectively simple if it is c.e. and its complement is effectively immune. Every effectively simple set is simple and Turing-complete.
- A set [math]\displaystyle{ I \subseteq \mathbb{N} }[/math] is called hyperimmune if [math]\displaystyle{ I }[/math] is infinite, but [math]\displaystyle{ p_I }[/math] is not computably dominated, where [math]\displaystyle{ p_I }[/math] is the list of members of [math]\displaystyle{ I }[/math] in order.[2]
- A set [math]\displaystyle{ S \subseteq \mathbb{N} }[/math] is called hypersimple if it is simple and its complement is hyperimmune.[3]
Notes
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.
- 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. https://archive.org/details/classicalrecursi0000odif.
- Nies, André (2009). Computability and randomness. Oxford Logic Guides. 51. Oxford: Oxford University Press. ISBN 978-0-19-923076-1.
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