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16
Enumeration Reducibility, Nondeterministic Computations and Relative . . .
 RECURSION THEORY WEEK, OBERWOLFACH 1989, VOLUME 1432 OF LECTURE NOTES IN MATHEMATICS
, 1990
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Embeddings into the Medvedev and Muchnik lattices of Π0 1 classes
 Archive for Mathematical Logic
, 2004
"... Archive for Mathematical Logic Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of nonempty Π 0 1 subsets of 2 ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are latticeembeddable below any nonzero ele ..."
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Cited by 22 (17 self)
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Archive for Mathematical Logic Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of nonempty Π 0 1 subsets of 2 ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are latticeembeddable below any nonzero element of Pw. We show that many countable distributive lattices are latticeembeddable below any nonzero element of PM. 1
Medvedev degrees of 2dimensional subshifts of finite type. Ergodic Theory and Dynamical Systems
"... In this paper we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new counterexample in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial recursive functionals from X into Y and vice versa. The Medvedev deg ..."
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Cited by 17 (9 self)
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In this paper we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new counterexample in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial recursive functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursiontheoretic literature on the lattice of Medvedev degrees of nonempty Π 0 1 subsets of {0, 1} N. This lattice is known as Ps. We prove that Ps consists precisely of the Medvedev degrees of 2dimensional subshifts of finite type. We use this result to obtain an infinite collection of 2dimensional subshifts of finite type which are, in a certain sense, mutually incompatible. Definition 1. Let A be a finite set of symbols. The full 2dimensional shift on A is the dynamical system consisting of the natural action of Z2 on the compact set AZ2. A 2dimensional subshift is a nonempty closed set X ⊆ AZ2 which is invariant under the action of Z2. A 2dimensional subshift X is said to be of finite type if it is defined by a finite set of forbidden configurations. An interesting paper on 2dimensional subshifts of finite type is Mozes [22]. A standard reference for the 1dimensional case is the book of Lind/Marcus [20], which also includes an appendix [20, §13.10] on the 2dimensional case.
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 find a ..."
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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.
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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Cited by 4 (3 self)
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Research supported by NSF grants DMS0600823 and DMS0652637.
Intermediate logics and factors of the Medvedev lattice
 Ann. Pure Appl. Logic
"... Abstract. We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them. 1. ..."
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Cited by 2 (1 self)
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Abstract. We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them. 1.
BINARY SUBTREES WITH FEW LABELED PATHS
"... Abstract. We prove several quantitative Ramseyan results involving ternary complete trees with {0, 1}labeled edges where we attempt to nd a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson's in computability theory; we show ..."
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Cited by 2 (1 self)
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Abstract. We prove several quantitative Ramseyan results involving ternary complete trees with {0, 1}labeled edges where we attempt to nd a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson's in computability theory; we show that there is a bounded Π 0 1 class of positive measure which is not strongly (Medvedev) reducible to DNR3; in fact, the class of 1random reals is not strongly reducible to DNR3. 1.
ON THE STRUCTURE OF THE MEDVEDEV LATTICE
, 2008
"... We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size 2 2 ℵ 0, the size of the lattice itself. We also prove that it is consis ..."
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Cited by 1 (0 self)
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We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size 2 2 ℵ 0, the size of the lattice itself. We also prove that it is consistent that the lattice has chains of size 2 2 ℵ 0, and in fact that these big chains occur in every interval that has a big antichain. We also study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice. 1
Computability of Countable Subshifts in One Dimension ⋆
, 2011
"... We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of CantorBendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 Z is is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In cont ..."
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We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of CantorBendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 Z is is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In contrast, effectively closed (Π 0 1) subshifts of rank three contain only computable elements, but Π 0 1 subshifts of rank four may contain members of arbitrary ∆ 0 2 degree. There is no subshift of rank ω + 1.
r.e. separating classes
"... Important examples of Π0 1 classes of functions f ∈ ωω are the classes of sets (elements of ω2) which separate a given pair of disjoint r.e. sets: S2(A0, A1): = { f ∈ ω2: (∀i < 2)(∀x ∈ Ai)f(x) � = i}. A wider class consists of the classes of functions f ∈ ωk which in a generalized sense separate ..."
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Important examples of Π0 1 classes of functions f ∈ ωω are the classes of sets (elements of ω2) which separate a given pair of disjoint r.e. sets: S2(A0, A1): = { f ∈ ω2: (∀i < 2)(∀x ∈ Ai)f(x) � = i}. A wider class consists of the classes of functions f ∈ ωk which in a generalized sense separate a ktuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: Sk(A0,..., Ak−1): = { f ∈ ωk: (∀i < k)(∀x ∈ Ai)f(x) � = i}. We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let Sm k denote the Medvedev degrees of those Sk(A0,..., Ak−1) such that no m + 1 sets among A0,..., Ak−1 have a nonempty intersection. It is shown that each Sm k is an upper semilattice but not a lattice. The degree of the set of kary diagonally nonrecursive functions DNRk is the greatest element of S1 k. If 2 ≤ l < k, then 0M is the only degree in S1 l which is below a member of S1 k. Each Sm k is densely ordered and has the splitting property and the same holds for the lattice Lm k it generates. The elements of Sm k are exactly the joins of elements of S1 i for ⌈ k ⌉ ≤ i ≤ k. m