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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 definabl ..."
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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 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.
2004], The 89theory of R( ; _; ^) is undecidable
 Trans. Am. Math. Soc
"... Abstract 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 long standing open question. A negative s ..."
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Cited by 2 (2 self)
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Abstract 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 long standing 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.
The ∀∃theory of R(≤, ∨, ∧) is undecidable
 Trans. Amer. Math. Soc
, 2004
"... Abstract. 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 ..."
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Abstract. 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. 1.
Abstract
"... We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 G  registers, and each of the 2 k subsets of the registers contributes some informa ..."
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We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 G  registers, and each of the 2 k subsets of the registers contributes some information. 1
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
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.
EXACT PAIRS FOR THE IDEAL OF THE KTRIVIAL SEQUENCES IN THE TURING DEGREES
, 2012
"... The Ktrivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later i ..."
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The Ktrivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later in [Nie09, Problem 5.5.8]. We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists a Ktrivial degree d such that for all degrees a, b which are not Ktrivial and a> d, b> d there exists a degree v which is not Ktrivial and a> v, b> v. This work sheds light to the question of the definability of the Ktrivial degrees in the c.e. degrees.
Interpreting N in the computably enumerable weak truth table degrees
"... We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an im ..."
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We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an important further problem is the question of the actual complexity of the theory. If the structure is arithmetical, then its theory can be interpreted in true arithmetic, i.e. Th(N; +; \Theta). Thus an upper bound is ; (!) , the complexity of Th(N; +; \Theta). Here an interpretation of theories is a manyone reduction based on a computable map defined on sentences in some natural way. An example of an arithmetical structure is D T ( ; 0 ), the Turing degrees of \Delta 0 2 sets. Shore [16] proved that true arithmetic can be interpreted in Th(D T ( ; 0 )). A stronger result is interpretability without parameters of a copy of (N; +; \Theta) in the structure (interpretability of struc...
Interpreting Arithmetic in the R.E. Degrees Under ...Induction
"... . We study the problem of the interpretability of arithmetic in the r.e. degrees in models of fragments of Peano arithmetic. The main result states that there is an interpretation # ## # # such that every formula # of Peano arithmetic corresponds to a formula # # in the language of the partial ..."
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. We study the problem of the interpretability of arithmetic in the r.e. degrees in models of fragments of Peano arithmetic. The main result states that there is an interpretation # ## # # such that every formula # of Peano arithmetic corresponds to a formula # # in the language of the partial ordering of r.e. degrees such that for every model N of # 4 induction, N = # if and only if RN = # # , where RN is the structure whose universe is the collection of r.e. degrees in N . This supplies, for example, statements #m about the r.e. degrees which are equivalent (over I# 4 ) to I#m for every m > 4. 1. Introduction. A basic goal of reverse mathematics is to determine the axiom systems needed to prove particular theorems of mathematics by showing that they are equivalent (over a given base theory) to some specific axiom system. (See [12] for a general introduction to reverse mathematics in the setting of second order arithmetic.) In reverse recursion theory our setti...