On Schnorr and computable randomness, martingales, and machines

Downey, Rodney G.; Griffiths, Evan J.; LaForte, Geoffrey

doi:10.1002/malq.200310121

Cited by 31 publications

(63 citation statements)

References 20 publications

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“…While all reasonable relativizations of Martin-Löf randomness coincide, this is not the case for other notions of randomness. For example, Miyabe [32], following work of Downey, Griffiths and LaForte [13], examined a partial relativization of Schnorr randomness with only truth- A -test, and so every set which is Demuth random relative to A is also Demuth BLR A -random. If A is computably dominated, then bounding by A-computable functions and bounding by computable functions are the same, and so the notions coincide.…”

Section: Introductionmentioning

confidence: 99%

Characterizing Lowness for Demuth Randomness

Bienvenu¹,

Downey²,

Greenberg³

et al. 2014

J. symb. log.

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Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

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Section: Introductionmentioning

confidence: 99%

Characterizing Lowness for Demuth Randomness

Bienvenu¹,

Downey²,

Greenberg³

et al. 2014

J. symb. log.

Self Cite

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“…real. This contrasted with the theorem of Downey, Griffiths and LaForte [20] who showed that if a left-c.e. real was Kurtz random, then its Turing degree must resemble the halting problem in that it must be high (i.e.…”

Section: Theorem 334 (Wang [103]) a Real α Is Kurtz Random Iff Thermentioning

confidence: 76%

“…Schnorr trivial reals behave quite differently than do Schnorr low reals and the Ktrivials. Downey and Griffiths constructed a Schnorr trivial real and Downey, Griffiths and LaForte [20] showed that they can even be Turing complete, though they do not occur in every computably enumerable Turing degree. Subsequently, they have been investigated by Johanna Franklin [31].…”

Section: Theorem 59 (Nies [76]) Suppose That a Is Low For Computablmentioning

confidence: 99%

Algorithmic randomness and computability

Downey¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…set A is autocomplex if and only if it is Turing equivalent to the halting problem (see Downey and Hirschfeldt [13, Theorem 8.16.7]). Downey, Griffiths and LaForte [12, Theorem 8] showed that there exists a c.e. Schnorr trivial set which is Turing equivalent to the halting problem.…”

Section: Proposition 5 Low (Sr Sr) ⊆ Low(mlr Wr)mentioning

confidence: 99%

Unified characterizations of lowness properties via Kolmogorov complexity

Kihara

Miyabe

2014

Arch. Math. Logic

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Consider a randomness notion C. A uniform test in the sense of C is a total computable procedure that each oracle X produces a test relative to X in the sense of C. We say that a binary sequence Y is C-random uniformly relative to X if Y passes all uniform C tests relative to X . Suppose now we have a pair of randomness notions C and D where C ⊆ D, for instance Martin-Löf randomness and Schnorr randomness. Several authors have characterized classes of the form Low(C, D) which consist of the oracles X that are so feeble that C ⊆ D X . Our goal is to do the same when the randomness notion D is relativized uniformly: denote by Low (C, D) the class of oracles X such that every C-random is uniformly D-random relative to X . (1) We show that X ∈ Low (MLR, SR) if and only if X is c.e. tt-traceable if and only if X is anticomplex if and only if X is Martin-Löf packing measure zero with respect to all computable dimension functions.(2) We also show that X ∈ Low (SR, WR) if and only if X is computably i.o. tt-traceable if and only if X is not totally complex if and only if X is Schnorr Hausdorff measure zero with respect to all computable dimension functions.

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On Schnorr and computable randomness, martingales, and machines

Abstract: We examine the randomness and triviality of reals using notions arising from martingales and prefix-free machines.

Cited by 31 publications

References 20 publications

Characterizing Lowness for Demuth Randomness

Characterizing Lowness for Demuth Randomness

Algorithmic randomness and computability

Unified characterizations of lowness properties via Kolmogorov complexity

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