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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...
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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Dynamic Properties of Computably Enumerable Sets
 In Computability, Enumerability, Unsolvability, volume 224 of London Math. Soc. Lecture Note Ser
, 1995
"... A set A ` ! is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let E denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating t ..."
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A set A ` ! is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let E denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating the denable (especially Edenable) properties of a c.e. set A to its iinformation contentj, namely its Turing degree, deg(A), under T , the usual Turing reducibility. [Turing 1939]. Recently, Harrington and Soare answered a question arising from Post's program by constructing a nonemptly Edenable property Q(A) which guarantees that A is incomplete (A !T K). The property Q(A) is of the form (9C)[A ae m C & Q \Gamma (A; C)], where A ae m C abbreviates that iA is a major subset of Cj, and Q \Gamma (A; C) contains the main ingredient for incompleteness. A dynamic property P (A), such as prompt simplicity, is one which is dened by considering how fast elements elements enter A relat...
Differences of Computably Enumerable Sets
, 1999
"... We consider the lower semilattice D of differences of c.e. sets under inclusion. It is shown that D is not distributive as a semilattice, and that the c.e. sets form a definable subclass. 1 Introduction A persistent open problem about the lattice E of computably enumerable (c.e.) sets under inclusi ..."
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We consider the lower semilattice D of differences of c.e. sets under inclusion. It is shown that D is not distributive as a semilattice, and that the c.e. sets form a definable subclass. 1 Introduction A persistent open problem about the lattice E of computably enumerable (c.e.) sets under inclusion is to determine the least number k such that the \Sigma k theory is undecidable. Lachlan [6] proved that the \Sigma 2 theory in the language of lattices is decidable, while one of the various known proofs of undecidability for Th(E), in that case due to Harrington, shows that in fact the \Sigma 8 theory in the language of lattices is undecidable (see [10], p. 381 for a sketch of that proof). Thus a very unsatisfying gap of 6 quantifier alternations remains. The reason why the undecidability proofs are so "bad" is that the coding used is very indirect. For instance, first one codes the class of finite symmetric graphs (which has an hereditarily undecidable \Sigma 2 theory) in the cl...
DECISION PROBLEM FOR SEPARATED DISTRIBUTIVE LATTICES
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