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21
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.
Mass problems and almost everywhere domination
 Mathematical Logic Quarterly
, 2007
"... We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED ..."
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Cited by 10 (7 self)
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We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω. Let 1 and 0 be the top and bottom elements of Pw. We show that inf(b1,1) and inf(b2,1) and inf(b3,1) belong to Pw and that 0 < inf(b1,1) < inf(b2,1) < inf(b3,1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, we show that inf(b1,1) and inf(b3,1) but not inf(b2,1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more selfcontained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan.
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.
RANDOMNESS NOTIONS AND PARTIAL RELATIVIZATION
"... Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the MartinLöf random reals relative to A. We discuss the connections ..."
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Cited by 2 (2 self)
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Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the MartinLöf random reals relative to A. We discuss the connections with LRreducibility and also study the reducibility associated with weak 2randomness. 1.
COMPUTABILITY, TRACEABILITY AND BEYOND
"... This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability ..."
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This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability, and show that the index set of the strongly jump traceable computably enumerable (c.e.) sets is Π0 4complete. This shows that the problem of deciding if a c.e. set is strongly jump traceable, is as hard as it can be. We define a strengthening of strong jump traceability, called hyper jump traceability, and prove some interesting results about this new class. Despite the fact that the hyper jump traceable sets have their origins in algorithmic randomness, we are able to show that they are natural examples of several Turing degree theoretic properties. For instance, we show that the hyper jump traceable sets are the first example of a lowness class with no promptly simple members. We also study the dual highness notions obtained from strong jump traceability, and explore their degree theoretic properties.
CHARACTERIZING THE STRONGLY JUMPTRACEABLE SETS VIA RANDOMNESS
"... Abstract. We show that if a set A is computable from every superlow 1random set, then A is strongly jumptraceable. Together with a result from [9], this theorem shows that the computably enumerable jumptraceable sets are exactly the computably enumerable sets computable from every superlow 1rand ..."
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Abstract. We show that if a set A is computable from every superlow 1random set, then A is strongly jumptraceable. Together with a result from [9], this theorem shows that the computably enumerable jumptraceable sets are exactly the computably enumerable sets computable from every superlow 1random set. We also prove the analogous result for superhighness: a c.e. set is strongly jumptraceable if and only if it is computable from any superhigh random set. Finally, we show that for each cost function c with the limit condition there is a random ∆ 0 2 set Y such that each c.e. set A �T Y obeys c. 1.
BJØRN KJOSHANSSEN AND ANDRÉ NIES
"... Abstract. We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets. 1. ..."
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Abstract. We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets. 1.
superhighness
"... We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets. ..."
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We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets.