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.
Formal definitions and some properties
- A set is called immune if is infinite, but for every index , we have . Or equivalently: there is no infinite subset of that is recursively enumerable.
- A set is called simple if it is recursively enumerable and its complement is immune.
- A set is called effectively immune if is infinite, but there exists a recursive function such that for every index , we have that .
- A set 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 is called hyperimmune if is infinite, but is not computably dominated, where is the list of members of in order.
- A set is called hypersimple if it is simple and its complement is hyperimmune.
- Nies (2009) p.35
- Nies (2009) p.27
- Nies (2009) p.37
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- 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.