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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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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
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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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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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.
Effectively Closed Sets and Enumerations
, 2007
"... An effectively closed set, or Π 0 1 class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable fu ..."
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An effectively closed set, or Π 0 1 class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of Π 0 1 classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of Π 0 1 classes and for the subclasses of decidable and of homogeneous Π 0 1 classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends.
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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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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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.
ON THE DEGREE SPECTRUM OF A Π01 CLASS
"... Abstract. For any P ⊆ 2ω, define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists A ∈ P of degree a. We prove a number of basic properties of the structure which is the degree spectra of Π01 classes ordered by inclusion and also study in detail some other ..."
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Abstract. For any P ⊆ 2ω, define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists A ∈ P of degree a. We prove a number of basic properties of the structure which is the degree spectra of Π01 classes ordered by inclusion and also study in detail some other phenomena relating to the study of Π01 classes from a degree theoretic point of view, which are brought to light as a result of this analysis. 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.