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Definability in the Enumeration Degrees
 Sacks Symposium
, 1997
"... We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the second order theory of arithmetic. By an e#ective version of coding lemma, we show that the first order theory ..."
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We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the second order theory of arithmetic. By an e#ective version of coding lemma, we show that the first order theory of the enumeration degrees of the # 0 2 sets is not decidable. 1 Introduction Definition 1.1 The following terms specify the variations on relative enumerability. . An recursive enumeration procedure U is a recursively enumerable collection of pairs #a, b# in which a and b are finite subsets of N. . U(A) is equal to B if B = {n : (#a)(#b)[a # A and #a, b# # U and n # b]} . . Say that B is enumeration reducible to A (A # e B) if there is a recursive enumeration procedure U such that U(A) = B. Similarly, A is enumeration equivalent to B (A # e B) if A # e B and B # e A. Definition 1.2 The enumeration degrees are defined as follows. . The enumeration degree of ...
Bounding and nonbounding minimal pairs in the enumeration degrees
 J. Symbolic Logic
"... Abstract. We show that every nonzero ∆ 0 2 edegree bounds a minimal pair. On the other hand, there exist Σ 0 2 edegrees which bound no minimal pair. 1. ..."
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Abstract. We show that every nonzero ∆ 0 2 edegree bounds a minimal pair. On the other hand, there exist Σ 0 2 edegrees which bound no minimal pair. 1.
Noncappable Enumeration Degrees Below 0
"... We prove that there exists a noncappable enumeration degree strictly below 0 0 e . Two notions of relative computability, Turing and enumeration reducibility, are basic to any natural finestructure theory for the classes of computable and incomputable objects. Of the theories for the correspondin ..."
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We prove that there exists a noncappable enumeration degree strictly below 0 0 e . Two notions of relative computability, Turing and enumeration reducibility, are basic to any natural finestructure theory for the classes of computable and incomputable objects. Of the theories for the corresponding degree structures (D D D and D D D e ), that for the Turing degrees is the better developed, mainly due to the depth of knowledge of specific local structure (see for example Lerman [Le83], Odifreddi [Od89] and Soare [So87]). Despite its importance (see [Co90]) in applications to nondeterministic computations, to relative computability involving partial information, in providing models of calculus, and in setting 1991 Mathematics Subject Classification. 03D30. Key words and phrases. Enumeration operator, enumeration degree, \Sigma 2 set. Research partially supported by British CouncilMURST grant no. ROM/889 /92/81, S.E.R.C. Research Grant no. GR/H 02165 and EC Human Capital and Mobili...
DISCONTINUOUS PHENOMENA
"... ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 1. ..."
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ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 1.
Cupping and Noncupping in the Enumeration Degrees of ... Sets
"... We prove the following three theorems on the enumeration degrees of # 0 2 sets. Theorem A: There exists a nonzero noncuppable # 0 2 enumeration degree. Theorem B: Every nonzero # 0 2 enumeration degree is cuppable to 0 # e by an incomplete total enumeration degree. Theorem C: There exists a nonzero ..."
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We prove the following three theorems on the enumeration degrees of # 0 2 sets. Theorem A: There exists a nonzero noncuppable # 0 2 enumeration degree. Theorem B: Every nonzero # 0 2 enumeration degree is cuppable to 0 # e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low # 0 2 enumeration degree with the anticupping property.
Enumeration Degrees Are Not Dense
, 1996
"... We show that the \Pi 0 2 enumeration degrees are not dense. This answers a question posed by Cooper. 1 Background Cooper [3] showed that the enumeration degrees (edegrees) are not dense. He also showed that the edegrees of the \Sigma 0 2 sets are dense [2]. Since Cooper's nondensity proof const ..."
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We show that the \Pi 0 2 enumeration degrees are not dense. This answers a question posed by Cooper. 1 Background Cooper [3] showed that the enumeration degrees (edegrees) are not dense. He also showed that the edegrees of the \Sigma 0 2 sets are dense [2]. Since Cooper's nondensity proof constructs a \Sigma 0 7 set, he posed the question: What is the least n (2 n 6) such that the edegrees below 0 n are not dense? (We consider the Turing degrees to be a subset of the edegrees by identifying the Turing degree of a set A with the edegree of A \Phi A.) Cooper also conjectured that the edegrees of the \Pi 0 2 sets are dense. Here we show that the edegrees are not dense in the \Pi 0 2 edegrees. Thus the answer to Cooper's question is 2 and his conjecture is false. This means that Cooper's proof that the \Sigma 0 2 edegrees are dense cannot be extended beyond the \Sigma 0 2 level in the arithmetical hierarchy. 2 Description of the Construction The basic plan of the cons...
Jumps of ...High EDegrees and Properly ... EDegrees
"... We show that the \Sigma 0 2 high edegrees coincide with the high edegrees. We also show that not every properly \Sigma 0 2 edegree is high. 1 Introduction Enumeration reducibility is the notion of relative enumerability of sets: a set A is enumeration reducible (or simply ereducible) to a set ..."
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We show that the \Sigma 0 2 high edegrees coincide with the high edegrees. We also show that not every properly \Sigma 0 2 edegree is high. 1 Introduction Enumeration reducibility is the notion of relative enumerability of sets: a set A is enumeration reducible (or simply ereducible) to a set B, in symbols, A e B, if there is an effective procedure for enumerating A given any enumeration of B. Formally, we define A e B if there is some computably enumerable set \Phi (called in this context an enumeration operator or simply an eoperator) such that A = fx : (9 finite D)[hx; Di 2 \Phi &D ` B]g (throughout the paper we identify finite sets with their canonical indices). We denote by j e the equivalence relation generated by the preordering relation e and deg e (A) denotes the equivalence class (or the edegree) of A. The partially ordered structure of the edegrees is denoted by D e ; its partial ordering is denoted by . D e is, in fact, an upper semilattice with least elem...
Jumps of ...High EDegrees and Properly
"... We show that the # 0 2 high edegrees coincide with the high edegrees. We also show that not every properly # 0 2 edegree is high. 1 Introduction Enumeration reducibility is the notion of relative enumerability of sets: a set A is enumeration reducible (or simply ereducible) to a set B, in sy ..."
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We show that the # 0 2 high edegrees coincide with the high edegrees. We also show that not every properly # 0 2 edegree is high. 1 Introduction Enumeration reducibility is the notion of relative enumerability of sets: a set A is enumeration reducible (or simply ereducible) to a set B, in symbols, A # e B, if there is an e#ective procedure for enumerating A given any enumeration of B. Formally, we define A # e B if there is some computably enumerable set # (called in this context an enumeration operator or simply an eoperator) such that A = {x : (# finite D)[#x, D# # #&D # B]} (throughout the paper we identify finite sets with their canonical indices). We denote by # e the equivalence relation generated by the preordering relation # e and deg e (A) denotes the equivalence class (or the edegree) of A. The partially ordered structure of the edegrees is denoted by D e ; its partial ordering is denoted by #. D e is, in fact, an upper semilattice with least el...
Structural Properties and ... Enumeration Degrees
"... We prove that each \Sigma 2 0 set which is hypersimple relative to ; 0 is noncuppable in the structure of the \Sigma 0 2 enumeration degrees. This gives a connection between properties of \Sigma 0 2 sets under inclusion and and the \Sigma 0 2 enumeration degrees. We also prove that some ..."
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We prove that each \Sigma 2 0 set which is hypersimple relative to ; 0 is noncuppable in the structure of the \Sigma 0 2 enumeration degrees. This gives a connection between properties of \Sigma 0 2 sets under inclusion and and the \Sigma 0 2 enumeration degrees. We also prove that some low nonc.e. enumeration degree contains no set which is simple relative to ; 0 . 1 Introduction There is a wide range of theorems in computability theory asserting that, in a certain degree structure R r of computably enumerable (c.e. ) sets under a reducibility r , a simplicity property of a c.e. set A implies the incompleteness of the rdegree of A. (Here a simplicity property requires that in some sense the complement of A is sparse.) An example of such a result is that a simple set cannot be bttcomplete ([Pos44]). While a simple set may be ttcomplete, the stronger notion of hypersimplicity of A even implies wttincompleteness. Downey and Jockusch [DJ87] showed that the wttdeg...
Splitting Properties of Total Enumeration Degrees
"... This paper (see Theorem 6 below) describes general conditions under which relative splittings and specified diamond embeddings are derivable in the local structure of the enumeration degrees. In so doing, three underlying themes are touched on  that of characterising the context for the Turing degr ..."
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This paper (see Theorem 6 below) describes general conditions under which relative splittings and specified diamond embeddings are derivable in the local structure of the enumeration degrees. In so doing, three underlying themes are touched on  that of characterising the context for the Turing degrees provided via enumeration reducibility; secondly, general questions of definability and the role of splitting and nonsplitting; and also (emerging from the techniques) the description of new relationships between information content and degree theoretic structure. Jockusch [1968] introduced the notion of a semirecursive set: A ⊆ ω is semirecursive if there is a computable f : ω² → ω for which, for all x, y: (i) f(x, y) = x or f(x, y) = y, (ii) x ∈ A ∨ y ∈ A ⇒ f(x, y) ∈ A. The paper begins with a result...