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25
Enumeration reducibility, nondeterministic computations and relative computability of partial functions
 in Recursion Theory Week, Proceedings Oberwolfach
, 1989
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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 24 (18 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 18 (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 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.
Noncomputable Spectral Sets
, 2007
"... iii For my Mama, whose *minimal index is computable (because it’s finite). ..."
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iii For my Mama, whose *minimal index is computable (because it’s finite).
Degrees of unsolvability of continuous functions
 Journal of Symbolic Logic
"... Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce ..."
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Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees nontotal. Several fundamental results are proved: a continuous function with nontotal degree has no least degree representation, settling a question asked by PourEl and Lempp; every noncomputable f ∈ C[0, 1] computes a noncomputable subset of N; there is a nontotal degree between Turing degrees a <T b iff b is a PA degree relative to a; S ⊆ 2N is a Scott set iff it is the collection of fcomputable subsets of N for some f ∈ C[0, 1] of nontotal degree; and there are computably incomparable f, g ∈ C[0, 1] which compute exactly the same subsets of N. Proofs draw from classical analysis and constructive analysis as well as from computability theory. §1. Introduction. The computable real numbers were introduced in Alan Turing’s famous 1936 paper, “On computable numbers, with an application to the Entscheidungsproblem ” [40]. Originally, they were defined to be the reals
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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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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Abstract. We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them. 1.
Algorithmic tests and randomness with respect to a class of measures
, 2011
"... This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the MartinLöf definition of randomness (with respect to computable measures) in term ..."
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Cited by 2 (1 self)
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This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the MartinLöf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli
Basic Subtoposes of the Effective Topos
, 2012
"... A fundamental concept in Topos Theory is the notion of subtopos: a subtopos of a topos E is a full subcategory which is closed under finite limits in E, and such that the inclusion functor has a left adjoint which preserves finite limits. It then follows that this subcategory is itself a topos, and ..."
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A fundamental concept in Topos Theory is the notion of subtopos: a subtopos of a topos E is a full subcategory which is closed under finite limits in E, and such that the inclusion functor has a left adjoint which preserves finite limits. It then follows that this subcategory is itself a topos, and its internal logic has