Difference between revisions of "Simple set"

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

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=34508
This article was adapted from an original article by S.N. Artemov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article