Archive for Mathematical Logic

, Volume 54, Issue 3–4, pp 329–358 | Cite as

Unified characterizations of lowness properties via Kolmogorov complexity

  • Takayuki Kihara
  • Kenshi MiyabeEmail author


Consider a randomness notion \({\mathcal{C}}\). A uniform test in the sense of \({\mathcal{C}}\) is a total computable procedure that each oracle X produces a test relative to X in the sense of \({\mathcal{C}}\). We say that a binary sequence Y is \({\mathcal{C}}\)-random uniformly relative to X if Y passes all uniform \({\mathcal{C}}\) tests relative to X. Suppose now we have a pair of randomness notions \({\mathcal{C}}\) and \({\mathcal{D}}\) where \({\mathcal{C} \subseteq \mathcal{D}}\), for instance Martin-Löf randomness and Schnorr randomness. Several authors have characterized classes of the form Low(\({\mathcal{C}, \mathcal{D}}\)) which consist of the oracles X that are so feeble that \({\mathcal{C} \subseteq \mathcal{D}^X}\). Our goal is to do the same when the randomness notion \({\mathcal{D}}\) is relativized uniformly: denote by Low \({\star(\mathcal{C},\mathcal{D})}\) the class of oracles X such that every \({\mathcal{C}}\)-random is uniformly \({\mathcal{D}}\)-random relative to X. (1) We show that \({X\in Low ^\star}\)(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 \({X\in Low^\star}\) (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.


Algorithmic randomness Lowness for randomness Uniform relativization Traceability Effective dimension 

Mathematics Subject Classification

03D32 68Q30 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan
  2. 2.School of Science and TechnologyMeiji UniversityKawasakiJapan

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