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Codable sets and orbits of computably enumerable sets

Published online by Cambridge University Press:  12 March 2014

Leo Harrington  and
Robert I. Soare
Leo Harrington
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA, E-mail: [email protected]
Robert I. Soare
Affiliation:
Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637, USA, E-mail: [email protected]
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Abstract

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let ε denote the structure of the computably enumerable sets under inclusion, ε = ({We}eϵω,⊆). We previously exhibited a first order ε-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets).

Here we show first that Q(X) implies that X has a certain “slowness” property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A ϵ ε there exists B in the orbit of A such that X ≤TB under relative Turing computability (≤T). We produce B using the -automorphism method we introduced earlier.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 1 - 28
Copyright
Copyright © Association for Symbolic Logic 1998

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