Schnorr Dimension

Downey, Rodney G.; Merkle, Wolfgang; Reimann, Jan

doi:10.1007/11494645_13

Cited by 4 publications

(4 citation statements)

References 15 publications

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“…Instead, we give the proof of the next theorem, a Schnorr version of Theorem 9. The following theorem generalizes a characterization of Schnorr dimension by Downey et al [17] (see also Downey and Hirschfeldt [13,Sect. 13.15]).…”

Section: Tot Anticomplex (Schnorr Trivial)supporting

confidence: 58%

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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Section: Tot Anticomplex (Schnorr Trivial)supporting

confidence: 58%

Unified characterizations of lowness properties via Kolmogorov complexity

Kihara

Miyabe

2014

Arch. Math. Logic

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“…A characterization of computable dimension in terms of computable machines has been obtained by Downey et al [1], and we conclude this section by a similar characterization in terms of decidable machines. In the context of time-bounded complexity, an equivalent formulation of the latter characterization has been previously demonstrated by Hitchcock [5], see Proposition 20.…”

Section: Proposition 14 (Downey and Griffiths)supporting

confidence: 64%

“…We give identical or very similar charaterizations of all three notions of randomness in terms of decidable machines; to the best of our knowledge, this is the first time that all three notions are characterized using a single type of Turing machine. Similary, we argue that the characterization of computable Hausdorff dimension in terms of computable machines due to Downey et al [1] extends to decidable machines. In Section 4, all the mentioned charaterizations are transferred to standard time-bounded Kolmogorov complexity by arguing that the latter is closely related to Kolmogorov complexity defined via decidable machines.…”

Section: Introductionsupporting

confidence: 68%

Reconciling Data Compression and Kolmogorov Complexity

Bienvenu

Merkle

Automata, Languages and Programming

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Abstract. While data compression and Kolmogorov complexity are both about effective coding of words, the two settings differ in the following respect. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of Kolmogorov complexity a word has many different codes and the minimum code for a given word cannot be found effectively. This gap is bridged by introducing decidable Turing machines and a corresponding notion of Kolmogorov complexity, where compressors and suitably normalized decidable machines are essentially the same concept. Kolmogorov complexity defined via decidable machines yields characterizations in terms of the intial segment complexity of sequences of the concepts of Martin-Löf randomness, Schnorr randomness, Kurtz randomness, and computable dimension. These results can also be reformulated in terms of time-bounded Kolmogorov complexity. Other applications of decidable machines are presented, such as a simplified proof of the Miller-Yu theorem (characterizing Martin-Löf randomness by the plain complexity of the initial segments) and a new characterization of computably traceable sequences via a natural lowness notion for decidable machines.

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“…Again it is possible to examine these concepts for stronger and weaker randomness notions such as Schnorr dimension. For instance, Downey, Merkle and Reimann [30] have shown that it is possible to have computably enumerable sets with nonzero Schnorr packing dimension, whereas their Schnorr Hausdorff dimension is 0. Much work remains to be done here with a plethora of open questions.…”

Section: Theorem 337 (Nies Stephan Terwijn [78]) Suppose That a Imentioning

confidence: 99%