Cone avoidance and randomness preservation

Simpson, Stephen G.; Stephan, Frank

doi:10.1016/j.apal.2015.03.001

Cited by 6 publications

(3 citation statements)

References 29 publications

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“…To define the notions of negligibility and non-negligibility, we need to review the definition of left-c.e. semi-measures, which were initially introduced by Solomonoff [29, 30] and first systematically studied by Levin and Zvonkin [20].…”

Section: Negligibility and Non-negligibilitymentioning

confidence: 99%

“…To prove Theorem 1.2, we prove a strengthening of Theorem 6.1 in terms of a family of weak notions of randomness; just as Theorem 1.1 follows from Theorem 6.1, so too will Theorem 1.2 follow from this strengthening. The following notion was explicitly defined by Higuchi et al [11] and was further studied by Simpson and Stephan [27]. Definition 6.7.…”

Section: A New Application Of the Techniquementioning

confidence: 99%

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Degrees of Randomized Computability

Hölzl¹,

Porter²

2021

Bull. symb. log

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In this survey we discuss work of Levin and V'yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V'yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections A and B, A is below B in this ordering if A \ B is negligible. The degree structure associated with this ordering, the Levin-V'yugin degrees (or LV-degrees), can be shown to be a Boolean algebra, and in fact a measure algebra.We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin-V'yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V'yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V'yugin's about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree. This is a ``preproof'' accepted article for The Bulletin of Symbolic Logic. This version may be subject to change during the production process.

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Section: Negligibility and Non-negligibilitymentioning

confidence: 99%

Section: A New Application Of the Techniquementioning

confidence: 99%

Degrees of Randomized Computability

Hölzl¹,

Porter²

2021

Bull. symb. log

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“…Kjos‐Hanssen, Merkle, and Stephan [12] showed that every

C

‐compression function is PA‐complete. Stephan and Simpson [19] gives yet another characterization of

K

‐trivials (reals of which all initial segments have the lowest possible Kolmogorov complexity) as the reals computable in every PA degree and relative to which the c.e. random real

normalΩ

is still 1‐random.…”

Section: Introductionmentioning

confidence: 99%

A computable analysis of majorizing martingales

Liu

2021

Bull. London Math. Soc.

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We give upper bounds for several highness properties in computability randomness theory. First, we prove that a certain discrete covering property (which requires to almost cover all sets in a certain class) does not imply the ability to compute a 1‐random real, answering a question of Greenberg, Miller and Nies. This also implies that an infinite set of incompressible strings does not necessarily compute a 1‐random real. Second, we prove that given a homogeneous binary tree that does not admit an infinite computable path, a sequence of bounded martingales whose initial capitals tend to zero (where a martingale is bounded if its range is a bounded set of reals), there exists a martingale S majorizing infinitely many of them such that S does not compute an infinite path of the tree. This implies that: (1) a certain highness notion (which degenerates non 1‐random into noncomputably random) does not imply PA‐completeness, answering a question of Miller; (2) the computably random reducibility is not equivalent to Turing reducibility, answering a question of Nies. The proof of the second result suggests that the coding power of the universal computably enumerable martingale lies in its infinite variance.

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