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Parameter Definability in the Recursively Enumerable Degrees
"... The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly defin ..."
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Cited by 40 (15 self)
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The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...
The ∀∃theory of R(≤, ∨, ∧) is undecidable
 TRANS. AMER. MATH. SOC
, 2004
"... The three quantifier theory of (R, ≤T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a longstanding open question. A negative solution ..."
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Cited by 4 (1 self)
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The three quantifier theory of (R, ≤T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a longstanding open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of R that lies between the two and three quantifier theories with ≤T but includes function symbols. Theorem. The two quantifier theory of (R, ≤, ∨, ∧), the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on R) is undecidable. The same result holds for various lattices of ideals of R which are natural extensions of R preserving join and infimum when it exits.
Parameter Definable Subsets of the Computably Enumerable Degrees
"... We prove definability results for the structure R T of computably enumerable Turing degrees. Some of the results can be viewed as approximations to an affirmative answer for the biinterpretability conjecture in parameters for. For instance, all uniformly computably enumerable sets of nonzero c.e ..."
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Cited by 2 (2 self)
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We prove definability results for the structure R T of computably enumerable Turing degrees. Some of the results can be viewed as approximations to an affirmative answer for the biinterpretability conjecture in parameters for. For instance, all uniformly computably enumerable sets of nonzero c.e. Turing degrees can be defined from parameters by a fixed formula. This implies that the finite subsets are uniformly definable. As a consequence we obtain a new ;definable ideal, and all arithmetical ideals are parameter definable. 1 Introduction Let R T denote the upper semilattice of computably enumerable (c.e.) Turing degrees. We are concerned with definability results in R T which can be viewed as approximations to the biinterpretability conjecture for R T . The biinterpretability conjecture in parameters for an arithmetical structure A (in brief, BIconjecture) states that there is a parameter defined copy M of (N; +; \Theta) and a parameter definable 11 map f : M 7! A. This h...
CONTENTS
"... 1 Introduction to computability theory 1 1.1 The basic concepts 2 1.1.1 Partial computable functions 2 1.1.2 Computably enumerable sets 5 ..."
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1 Introduction to computability theory 1 1.1 The basic concepts 2 1.1.1 Partial computable functions 2 1.1.2 Computably enumerable sets 5