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A Nonlow2 C.E. Degree Which Bounds No Diamond Bases
 Analysis of Information Utilisation (AUI). International Journal of HumanComputer Interaction
"... A computably enumerable (c.e.) Turing degree is a diamond base if and only if it is the bottom of a diamond of c.e. degrees with top 0 # . Cooper and Li [3] showed that no low 2 c.e. degree can bound a diamond bases. In the present paper, we show that there exists a nonlow 2 c.e. degree which do ..."
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A computably enumerable (c.e.) Turing degree is a diamond base if and only if it is the bottom of a diamond of c.e. degrees with top 0 # . Cooper and Li [3] showed that no low 2 c.e. degree can bound a diamond bases. In the present paper, we show that there exists a nonlow 2 c.e. degree which does not bound a diamond base. Thus, we refute an attractive natural attempt to define the jump class low 2 .
The ∀∃ theory of D(≤, ∨, ′ ) is undecidable
 In Proceedings of Logic Colloquium
, 2003
"... We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. ..."
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We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.
The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
 Memoirs of the American Mathematical Society
"... Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In ..."
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Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of αfiniteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and nondistributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the αr.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the firstorder language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of αr.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the αr.e. degrees is complicated, we get that for every admissible ordinal
A join theorem for the computably enumerable degrees
 Trans. Amer. Math. Soc
, 2004
"... Abstract. It is shown that for any computably enumerable (c.e.) degree w, if w � = 0, then there is a c.e. degree a such that (a ∨ w) ′ = a ′ ′ = 0 ′′ (so a is low2 and a ∨ w is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low2 c.e. degrees ..."
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Abstract. It is shown that for any computably enumerable (c.e.) degree w, if w � = 0, then there is a c.e. degree a such that (a ∨ w) ′ = a ′ ′ = 0 ′′ (so a is low2 and a ∨ w is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low2 c.e. degrees are not elementarily equivalent as partial orderings. 1.
Differences between Resource Bounded Degree Structures
"... We exhibit a structural difference between the truthtable degrees of the sets which are truthtable above 0 # and the PTIMETuring degrees of all sets. ..."
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We exhibit a structural difference between the truthtable degrees of the sets which are truthtable above 0 # and the PTIMETuring degrees of all sets.
ELEMENTARY DIFFERENCES AMONG FINITE LEVELS OF THE ERSHOV HIERARCHY
"... Abstract. We study the differences among finite levels of the Ershov hierarchies. We also give a brief survey on the current state of this area. Some questions are raised. 1. Preliminary Putnam [9] is the first one who introduced the nr.e. sets. Definition 1.1. (i) A set A is nr.e. if there is a r ..."
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Abstract. We study the differences among finite levels of the Ershov hierarchies. We also give a brief survey on the current state of this area. Some questions are raised. 1. Preliminary Putnam [9] is the first one who introduced the nr.e. sets. Definition 1.1. (i) A set A is nr.e. if there is a recursive function f: ω × ω → ω so that for each m, – f(0, m) = 0. – A(m) = lims f(s, m). – {sf(s + 1, m) � = f(s, m)}  ≤ n. • A Turing degree is nr.e. if it contains an nr.e. set. We use Dn to denote the collection of nr.e. degrees. For simplicity, we redefine D0 = D1 which is a little unusual. For other recursion notations, please refer to Soare [13]. In this paper, we work in the partially ordered language, L(≤), through out. L(≤) includes variables a, b, c, x, y, z,... and a binary relation ≤ intended to denote a partial order. Atomic formulas are x = y, x ≤ y. Σ0 formulas are built by the following induction definition. • Each atomic formula is Σ0. • ¬ψ for some Σ0 formula ψ. • ψ1 ∨ ψ2 for two Σ0 formula ψ1, ψ2. • ψ1 ∧ ψ2 for two Σ0 formula ψ1, ψ2 • ψ1 = ⇒ ψ2 for two Σ0 formula ψ1, ψ2. A formula ϕ is Σ1 if it is of the form ∃x1∃x2...∃xnψ(x1, x2,..., xn) for some Σ0 formula ψ. For each n ≥ 1, a formula ϕ is Πn if it is the form ¬ψ for some Σn formula ψ and a formula ϕ is Σn+1 if it is the form ∃x1∃x2...∃xmψ(x1, x2,..., xm) for some Πn formula ψ. A sentence is a formula without free variables.
A rigid cone in the truthtable degrees with jump
, 2004
"... Each automorphism of the truthtable degrees with order and jump is equal to the identity on a cone with base 0 (4). A degree structure is said to be rigid on a cone if each automorphism of the structure is equal to the identity on the set of degrees above a fixed degree. It is known [4] that the st ..."
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Each automorphism of the truthtable degrees with order and jump is equal to the identity on a cone with base 0 (4). A degree structure is said to be rigid on a cone if each automorphism of the structure is equal to the identity on the set of degrees above a fixed degree. It is known [4] that the structure of the Turing degrees with jump is rigid on a cone. This is shown by applying a jump inversion theorem and results on initial segments. In this paper it is shown using a weaker jump inversion theorem that also the structure of truthtable degrees with jump is rigid on a cone. For definitions relating to initial segments we refer to [13]. 1 Initial segment construction Lemma 1.1. Suppose for each e, g lies on a tree Te which is esplitting for some c for some tables with the properties of Proposition 4.9, in the sense of [5]. Then g is hyperimmunefree. Proof. For each e ∈ ω there exists e ∗ ∈ ω such that for all stages s and all oracles g, if {e ∗ } g s(x) ↓ then {e ∗ } g (x) = {e} g (x) and {e} g s(y) ↓ for all y ≤ x. If g lies on Te ∗ then it follows that {e}g is total and {e ∗ } T (σ) (x) ↓ for each σ of length x + 1. Hence {e} g = {e ∗ } g is dominated by the recursive function f(x) = max{{e} T (σ) (x) : σ  = x + 1}. Proposition 1.2. Let L be a Σ 0 4(y)presentable upper semilattice with least and greatest element. Then there exist t, i, g such that 1. t: ω → 2 is 0 ′ ′computable, 2. i is the characteristic function of a set I such that I ≤m y (3), 3. g ′ ′ (e) = t(i(0),..., i(e)) for all e ∈ ω, 4. [0, g] is isomorphic to L, and 5. g is hyperimmunefree 1 Proof. The proof in [5] must be modified to employ the lattice tables of Proposition
Undecidability and 1types in intervals of the computably enumerable degrees
 Ann. Pure Appl. Logic
, 2000
"... We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1types. subject code classifications: 03D25 (03C65 03D35 06A06) ..."
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We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1types. subject code classifications: 03D25 (03C65 03D35 06A06)