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Conference on Computability in Europe

CiE 2019: Computing with Foresight and Industry pp 50–61Cite as

Uniform Relativization

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

Abstract

This paper is a tutorial on uniform relativization. The usual relativization considers computation using an oracle, and the computation may not work for other oracles, which is similar to Turing reduction. The uniform relativization also considers computation using oracles, however, the computation should work for all oracles, which is similar to truth-table reduction. The distinction between these relativizations is important when we relativize randomness notions in algorithmic randomness, especially Schnorr randomness. For Martin-Löf randomness, its usual relativization and uniform relativization are the same so we do not need to care about this uniform relativization.

We focus on two specific examples of uniform relativization: van Lambalgen’s theorem and lowness. Van Lambalgen’s theorem holds for Schnorr randomness with the uniform relativization, but not with the usual relativization. Schnorr triviality is equivalent to lowness for Schnorr randomness with the uniform relativization, but not with the usual relativization. We also discuss some related known results.

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Notes

  1. 1.

    Every noncomputable c.e. Turing degree contains a hypersimple set [26, Proposition III.3.13] while a hypersimple set is not tt-complete [26, Theorem III.3.10].

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Acknowledgements

The author thanks the anonymous reviewers for their careful reading of the manuscript and their suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Meiji University, Kawasaki, Kanagawa, Japan

    Kenshi Miyabe

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  1. Kenshi Miyabe
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Correspondence to Kenshi Miyabe .

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Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

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Miyabe, K. (2019). Uniform Relativization. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_5

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_5

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