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15
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.
Isomorphisms Of Splits Of Computably Enumerable Sets
 J. OF SYMBOLIC LOGIC
, 2002
"... We show that if A and A are automorphic via # then the structures SR (A) and SR ( 3 isomorphic via an isomorphism # induced by #. Then we use this result to classify completely the orbits of hhsimple sets. ..."
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Cited by 6 (6 self)
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We show that if A and A are automorphic via # then the structures SR (A) and SR ( 3 isomorphic via an isomorphism # induced by #. Then we use this result to classify completely the orbits of hhsimple sets.
On the Orbits of Computable Enumerable Sets
, 2007
"... The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; th ..."
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Cited by 6 (5 self)
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The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof).
The complexity of orbits of computably enumerable sets
 BULLETIN OF SYMBOLIC LOGIC
, 2008
"... The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; ..."
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Cited by 3 (1 self)
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The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof).
Extending and Interpreting Post’s Programme
, 2008
"... Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic in ..."
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Cited by 2 (2 self)
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Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals which produced richness of local computabilitytheoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Posts programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Posts programme, and look at two interesting developments of Posts approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computabilitytheoretic infrastructure. We will conclude by viewing Posts programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world.
Invariance in E ∗ and EΠ
 Trans. Amer. Math. Soc
"... Abstract. We define G, a substructure of EΠ (the lattice of Π 0 1 classes) and show that a quotient structure of G, G ♦ , is isomorphic to E ∗. The result builds on the ∆ 0 3 isomorphism machinery, and allows us to transfer invariant classes from E ∗ to EΠ, though not, in general, orbits. Further pr ..."
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Abstract. We define G, a substructure of EΠ (the lattice of Π 0 1 classes) and show that a quotient structure of G, G ♦ , is isomorphic to E ∗. The result builds on the ∆ 0 3 isomorphism machinery, and allows us to transfer invariant classes from E ∗ to EΠ, though not, in general, orbits. Further properties of G ♦ and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance. 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
Extensions, Automorphisms, and Definability
 CONTEMPORARY MATHEMATICS
"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."
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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here