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17
Some fundamental issues concerning degrees of unsolvability
 In [6], 2005. Preprint
, 2007
"... Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to f ..."
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Cited by 9 (8 self)
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Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, corecursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.
Totally ωcomputably enumerable degrees and bounding critical triples, preprint
"... Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1. ..."
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Cited by 8 (4 self)
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Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1.
MASS PROBLEMS ASSOCIATED WITH EFFECTIVELY CLOSED SETS
, 2011
"... earlier draft of this paper. The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov’s nonrigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of none ..."
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Cited by 7 (2 self)
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earlier draft of this paper. The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov’s nonrigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let Ew be this lattice. We show that Ew provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in Ew which are associated with such problems. In addition, we present some structural results concerning the lattice Ew. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how Ew can be applied in symbolic dynamics, toward the classification of tiling problems
Invariance in E ∗ and EΠ
 Trans. Amer. Math. Soc
"... Abstract. We define G, a substructure of EΠ (the lattice of Π 0 1 classes) and show that a quotient structure of G, G ♦ , is isomorphic to E ∗. The result builds on the ∆ 0 3 isomorphism machinery, and allows us to transfer invariant classes from E ∗ to EΠ, though not, in general, orbits. Further pr ..."
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Cited by 2 (0 self)
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Abstract. We define G, a substructure of EΠ (the lattice of Π 0 1 classes) and show that a quotient structure of G, G ♦ , is isomorphic to E ∗. The result builds on the ∆ 0 3 isomorphism machinery, and allows us to transfer invariant classes from E ∗ to EΠ, though not, in general, orbits. Further properties of G ♦ and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance. 1.
Slender classes
, 2006
"... A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is ..."
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Cited by 2 (1 self)
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A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such think classes are automorphic in the lattice of Π 0 1 classes under inclusion. From this it follows that if the boolean algebra has a finite number n of atoms then the resulting classes are all automorphic. We prove a conjecture of Cholak and Downey [8] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class.
PROMPT SIMPLICITY, ARRAY COMPUTABILITY AND CUPPING
"... Abstract. We show that the class of c.e. degrees that can be joined to 0 ′ by an array computable c.e. degree properly contains the class of promptly simple degrees. 1. ..."
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Abstract. We show that the class of c.e. degrees that can be joined to 0 ′ by an array computable c.e. degree properly contains the class of promptly simple degrees. 1.
THE UPWARD CLOSURE OF A PERFECT THIN CLASS
"... Abstract. There is a perfect thin Π01 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2random degree.) Thus, in the Muchnik lattice of Π01 classes, the degree of 2random reals is comparable with the degree of some perfect thin class. This solves a quest ..."
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Abstract. There is a perfect thin Π01 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2random degree.) Thus, in the Muchnik lattice of Π01 classes, the degree of 2random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [16]. 1.
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.