Kolmogorov-Loveland Randomness and Stochasticity

Merkle, Wolfgang; Miller, Joseph S.; Nies, André; Reimann, Jan; Stephan, Frank

doi:10.1007/978-3-540-31856-9_35

Cited by 19 publications

(29 citation statements)

References 12 publications

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“…We give the formal definition, but refer to [31] for more details. An assignment is a (finite or infinite) sequence…”

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confidence: 99%

“…Thus the tests are pairs (S, M ), where S is a scan rule and M is a partial computable martingale; (S, M ) succeeds on Z if M succeeds on the bit sequence of σ Z S . (Equivalently, one may replace M by a stake function defined on the domain of S rather than on strings, see [31].) Given a test (S, M ), there are two tests where the scan rule and the martingale are total (even primitive recursive) [30], so that one of the two succeeds wherever (S, M ) succeeds.…”

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confidence: 99%

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Randomness and Computability: Open Questions

Miller

Nies

2006

Bull. symb. log.

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It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real.

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“…We give the formal definition, but refer to [31] for more details. An assignment is a (finite or infinite) sequence…”

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confidence: 99%

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confidence: 99%

Randomness and Computability: Open Questions

Miller

Nies

2006

Bull. symb. log.

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“…In contrast, van Lambalgen's theorem with the usual relativization does not hold for Schnorr randomness, computable randomness [37,55], Kurtz randomness [19] or weak 2-randomness [1].…”

Section: Uniform Relativizationmentioning

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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“…We also consider whether this is true for Kolmogorov-Loveland randomness, and relate this to a statistical question. Kolmogorov-Loveland randomness and stochasticity has been studied by, among others, Merkle [10], Merkle et al [11], and Bienvenu [1].…”

Section: Kolmogorov-loveland Stochasticity and Randomnessmentioning

confidence: 99%

How Much Randomness Is Needed for Statistics?

Kjos-Hanssen

Taveneaux

Thapen

2012

Lecture Notes in Computer Science

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In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure λ, a choice needs to be made. One approach is to allow randomness tests to access the measure λ as an oracle (which we call the "classical approach"). The other approach is the opposite one, where the randomness tests are completely effective and do not have access to the information contained in λ (we call this approach "Hippocratic"). While the Hippocratic approach is in general much more restrictive, there are cases where the two coincide. The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-Löf randomness and the measure λ is a Bernoulli measure, classical randomness and Hippocratic randomness coincide. In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity.

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Kolmogorov-Loveland Randomness and Stochasticity

Cited by 19 publications

References 12 publications

Randomness and Computability: Open Questions

Randomness and Computability: Open Questions

Unified characterizations of lowness properties via Kolmogorov complexity

How Much Randomness Is Needed for Statistics?

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