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37
Post's Program and incomplete recursively enumerable sets
, 1991
"... : A set A of nonnegative integers is recursively enumerable (r.e.) if A can be computably listed. It is shown that there is a first order property, Q(X), definable in E, the lattice of r.e. sets under inclusion, such that: (1) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing i ..."
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Cited by 26 (4 self)
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: A set A of nonnegative integers is recursively enumerable (r.e.) if A can be computably listed. It is shown that there is a first order property, Q(X), definable in E, the lattice of r.e. sets under inclusion, such that: (1) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing incomplete; and (2) there exists an r.e. set A satisfying Q(A). This resolves a long open question stemming from Post's Program of 1944, and it sheds new light on the fundamental problem of the relationship between the algebraic structure of an r.e. set A and the (Turing) degree of information which A encodes. Recursively enumerable (r.e.) sets have been a central topic in mathematical logic, in recursion theory (i.e. computability theory), and in undecidable problems. They are the next most effective type of set beyond recursive (i.e. computable) sets, and they occur naturally in many branches of mathematics. This together with the existence of nonrecursive r.e. sets has enabled them to pl...
Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
 Trans. Amer. Math. Soc
"... Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, o ..."
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Cited by 17 (6 self)
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Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.
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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Cited by 12 (5 self)
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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...
Asymptotic density and computably enumerable sets
"... We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in ..."
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Cited by 11 (6 self)
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We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: (i) The degrees of such sets A are precisely the nonlow c.e. degrees. (ii) There is a c.e. set A of density 1 with no computable subset of nonzero density. (iii) There is a c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A \ B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of “computable at density r ” where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness.
Definability, Automorphisms, And Dynamic Properties Of Computably Enumerable Sets
 this Bulletin
, 1996
"... . We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly ..."
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Cited by 9 (2 self)
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. We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility). 1. Introduction. All functions are on the nonnegative integers, # = {0, 1, 2, . . . }, and all sets will be subsets of #. Turing and G odel informally called a function computable if it can be calculated by a mechanical procedure, and regarded this as being synonymous with being specified by an "algorithm" or a "finite combinatorial procedure." They each formalized it as follows. 1 A function is Turing computable if it is definable by a Turing machine, as defined by Turing ...
On the definability of the double jump in the computably enumerable sets
 J. MATH. LOG
, 2002
"... We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: Let C = {a: a is the Turing degree of a � 0 3 set J ≥T 0 ′ ′}. Let D ⊆ C such that D is upward closed in C. Then there is an L(A) property ϕD(A) such that F ′ ′ ∈ D iff there is an A where A ..."
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Cited by 9 (5 self)
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We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: Let C = {a: a is the Turing degree of a � 0 3 set J ≥T 0 ′ ′}. Let D ⊆ C such that D is upward closed in C. Then there is an L(A) property ϕD(A) such that F ′ ′ ∈ D iff there is an A where A ≡T F and ϕD(A). A corollary of this is that, for all n ≥ 2, the highn (lown) computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin’s Invariance Conjecture.
There is no Fat Orbit
 Ann. Pure Appl. Logic
"... We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions. ..."
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Cited by 6 (3 self)
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We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.
Extension theorems, orbits, and automorphisms of the computably enumerable sets
 TRANS. AMER. MATH. SOC.
, 2008
"... We prove an algebraic extension theorem for the computably enumerable sets, E. Using this extension theorem and other work we then show if A and � A are automorphic via Ψ, then they are automorphic via Λ where Λ ↾ L ∗ (A) =ΨandΛ↾E ∗ (A) is∆0 3. We give an algebraic description of when an arbitrary ..."
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Cited by 6 (6 self)
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We prove an algebraic extension theorem for the computably enumerable sets, E. Using this extension theorem and other work we then show if A and � A are automorphic via Ψ, then they are automorphic via Λ where Λ ↾ L ∗ (A) =ΨandΛ↾E ∗ (A) is∆0 3. We give an algebraic description of when an arbitrary set �A is in the orbit of a computably enumerable set A. We construct the first example of a definable orbit which is not a ∆0 3 orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if A is simple and �A is in the same orbit as A, then they are in the same ∆0 6orbit and, furthermore, we provide a classification of when two simple sets are in the same orbit.