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24
Mass problems and randomness
 Bulletin of Symbolic Logic
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P.WesaythatP is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees ..."
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Cited by 37 (18 self)
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A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P.WesaythatP is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices Pw and Ps of weak and strong degrees of mass problems given by nonempty Π 0 1 subsets of 2 ω.Usingan abstract Gödel/Rosser incompleteness property, we characterize the Π 0 1 subsets of 2 ω whose associated mass problems are of top degree in Pw and Ps, respectively. Let R be the set of Turing oracles which are random in the sense of MartinLøf, and let r be the weak degree of R. We show that r is a natural intermediate degree within Pw. Namely, we characterize r as the unique largest weak degree of a Π 0 1 subset of 2 ω of positive measure.
Mass problems and hyperarithmeticity
, 2006
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of we ..."
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Cited by 24 (16 self)
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A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω CK 1 let hα be the weak degree of 0 (α) , the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p ∗ be the weak degree
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
An extension of the recursively enumerable Turing degrees
 Journal of the London Mathematical Society
, 2006
"... Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overco ..."
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Cited by 22 (16 self)
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Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with nonempty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally nonrecursive functions, and rn is the weak degree of the nrandom reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is onetoone, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 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 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 6 (1 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
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.
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