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Codable Sets and Orbits of Computably Enumerable Sets
 J. Symbolic Logic
, 1995
"... 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 E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order ..."
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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 E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order Edefinable 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 2 E there exists B in the orbit of A such that X T B under relative Turing computability ( T ). We produce B using the \Delta 0 3 automorphism method we introduced earli...
Highness and bounding minimal pairs
 Math. Logic Quarterly
, 1993
"... We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high2 nonbounding r.e. degree. 0 ..."
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We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high2 nonbounding r.e. degree. 0
Immunity Properties and the nC.E. Hierarchy
 in Theory and Applications of Models of Computation, Third International Conference on Computation and Logic, TAMC 2006, Beijing, May 2006, Proceedings, (JinYi
"... Abstract. We extend Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets, and characterise, in the spirit of Post [9], the degrees of the immune and hyperimmune d.c.e. sets. We also show that no properly d.c.e. set can be hhimmune, and indicate how to generalise these results to nc ..."
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Abstract. We extend Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets, and characterise, in the spirit of Post [9], the degrees of the immune and hyperimmune d.c.e. sets. We also show that no properly d.c.e. set can be hhimmune, and indicate how to generalise these results to nc.e. sets, n> 2. 1
Definability and Automorphisms of the Computably Enumerable Sets
, 2010
"... The computably enumerable (c.e.) sets have been central to computability theory since its inception. We study the structure of the c.e. sets, which forms a lattice E under set inclusion. Jump classes, such as the low degrees, allow us to classify the c.e. sets according to their information content. ..."
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The computably enumerable (c.e.) sets have been central to computability theory since its inception. We study the structure of the c.e. sets, which forms a lattice E under set inclusion. Jump classes, such as the low degrees, allow us to classify the c.e. sets according to their information content. The upward closed jump classes Ln and Hn have all been shown to be definable by a latticetheoretic formula, except for L1, the nonlow degrees, which is the only jump class whose definability was unknown. We say a class of c.e. degrees is invariant if it is the set of degrees of a class of c.e. sets that is invariant under automorphisms of E. All definable classes of degrees are invariant. We show that L1 is in fact noninvariant, thus proving a 1996 conjecture of Harrington and Soare in [3] that the nonlow degrees are not definable, and completing the problem of determining the definability of each jump class. 1