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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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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.
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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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.
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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Cited by 1 (1 self)
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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.
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.
Table of Contents
, 2006
"... The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology. ii ..."
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The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology. ii
DEGREES AND REVERSE MATHEMATICS
, 2011
"... We investigate the complexity of mathematical problems from two perspectives: Medvedev degrees and reverse mathematics. In the Medvedev degrees, we calculate the complexity of its firstorder theory, and we also calculate the complexities of the firstorder theories of several related structures. We ..."
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We investigate the complexity of mathematical problems from two perspectives: Medvedev degrees and reverse mathematics. In the Medvedev degrees, we calculate the complexity of its firstorder theory, and we also calculate the complexities of the firstorder theories of several related structures. We characterize the joinirreducible Medvedev degrees and deduce several consequences for the interpretation of propositional logic in the Medvedev degrees. We equate the size of chains of Medvedev degrees with the size of chains of sets of reals under ⊆. In reverse mathematics, we analyze the strength of classical theorems of finite graph theory generalized to the countable. In particular, we consider Menger’s theorem, Birkhoff’s theorem, and unfriendly partitions. BIOGRAPHICAL SKETCH Paul was born on February 28, 1983 in Richland, Washington during the final episode Goodbye, Farewell and Amen of the popular television series M*A*S*H.
1 Introduction Degrees of Unsolvability
, 2006
"... Modern computability theory began with Turing [Turing, 1936], where he introduced ..."
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Modern computability theory began with Turing [Turing, 1936], where he introduced
RESEARCH STATEMENT
, 2010
"... I am interested in studying the complexity of mathematical practice. In mathematics, as we all know, some structures are more complicated than others, some constructions more complicated than others, and some proofs more complicated than others. I am interested in understanding how to measure this c ..."
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I am interested in studying the complexity of mathematical practice. In mathematics, as we all know, some structures are more complicated than others, some constructions more complicated than others, and some proofs more complicated than others. I am interested in understanding how to measure this complexity and in measuring it. The motivations for this come from different areas. Form a foundational viewpoint, we want to know what assumptions we really need to do mathematics (ZF C is way much more than we usually use), and we are also interested in knowing what assumptions are used in the different areas of mathematics. Form a computational viewpoint, it is important to know what part of mathematics can be done by mechanical algorithms, and, even for the part that can’t be done mechanically, we want to know how constructive are the objects we deal with. Furthermore, it is sometimes the case that this computational analysis allows us to find connections between constructions in different areas of mathematics, and in many cases to obtain a deeper understanding of mathematical objects being analyzed. My work is quite diverse in terms of the techniques I have used, the approaches I have taken, and the areas of mathematics that I have analyzed. However, my background area is Computability Theory, and most of my work can be considered as part of this branch of Mathematical Logic.