Selection by Recursively Enumerable Sets

Abstract

For given sets A, B and Z of natural numbers where the members of Z are z ₀, z ₁, … in ascending order, one says that A is selected from B by Z if A(i) = B(z _i ) for all i. Furthermore, say that A is selected from B if A is selected from B by some recursively enumerable set, and that A is selected from B in n steps iff there are sets E ₀,E ₁,…,E _n such that E ₀ = A, E _n  = B, and E _i is selected from E _i + 1 for each i < n.

The following results on selections are obtained in the present paper. A set is ω-r.e. if and only if it can be selected from a recursive set in finitely many steps if and only if it can be selected from a recursive set in two steps. There is some Martin-Löf random set from which any ω-r.e. set can be selected in at most two steps, whereas no recursive set can be selected from a Martin-Löf random set in one step. Moreover, all sets selected from Chaitin’s Ω in finitely many steps are Martin-Löf random.

F. Stephan is supported in part by NUS grant R252-000-420-112. Part of the work was done while W. Merkle, F. Stephan and Y. Yang visited W. Wang at the Sun Yat-Sen University.