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Unified characterizations of lowness properties via Kolmogorov complexity

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Abstract

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.

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Authors and Affiliations

  1. School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, 923-1292, Japan

    Takayuki Kihara

  2. School of Science and Technology, Meiji University, Kawasaki, Kanagawa, 214-8571, Japan

    Kenshi Miyabe

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  1. Takayuki Kihara
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  2. Kenshi Miyabe
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Kihara, T., Miyabe, K. Unified characterizations of lowness properties via Kolmogorov complexity. Arch. Math. Logic 54, 329–358 (2015). https://doi.org/10.1007/s00153-014-0413-8

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