Mass problems and almost everywhere domination

Simpson, Stephen G.

doi:10.1002/malq.200710013

Cited by 13 publications

(3 citation statements)

References 21 publications

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“…Other interesting degrees in E w are closely related to other interesting topics in the foundations of mathematics. Among these topics are reverse mathematics [30,33], almost everywhere domination [36], measuretheoretic regularity [38], the hyperarithmetical hierarchy [10,38], effective Hausdorff dimension [19,40] and Kolmogorov complexity [15]. Proof.…”

Section: ] Andmentioning

confidence: 99%

Medvedev degrees of two-dimensional subshifts of finite type

Simpson

2012

Ergod. Th. Dynam. Sys.

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In this paper, we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial computable 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 recursion-theoretic literature on the lattices ℰs and ℰw of Medvedev degrees and Muchnik degrees of non-empty effectively closed subsets of {0,1}ℕ. We now prove that ℰs and ℰwconsist precisely of the Medvedev degrees and Muchnik degrees of two-dimensional subshifts of finite type. We apply this result to obtain an infinite collection of two-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.

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Section: ] Andmentioning

confidence: 99%

Medvedev degrees of two-dimensional subshifts of finite type

Simpson

2012

Ergod. Th. Dynam. Sys.

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“…We let JTH denote the class of sets that are h-JT-hard for some (computable) order function h. Every LR-hard set is in JTH via an op2 n q order function by [48] (or see [40, 8.4.15]).…”

Section: Diamond Classes and Ml-reducibilitymentioning

confidence: 99%

Coherent randomness tests and computing the $K$-trivial sets

Bienvenu¹,

Greenberg²,

Kučera³

et al. 2016

J. Eur. Math. Soc.

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We show that a Martin-Löf random set for which the effective version of the Lebesgue density theorem fails computes every K-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question 4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12 (3): 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all K-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every K-trivial set is computed by the two halves of a random set.The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a "smart" K-trivial set, all randoms from whom this set is computed have to compute all K-trivial sets.

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“…3] and inf(b α , 1) for all ordinal numbers α < ω CK 1 [66,68] belong to E w . In particular, the Muchnik degrees inf(r 2 , 1) [65, Sect.…”

Section: The Lattices E W and S Wmentioning

confidence: 99%