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A Church-Turing Thesis for Randomness?

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Connecting with Computability (CiE 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12813))

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Abstract

We discuss the difficulties in stating an analogue of the Church-Turing thesis for algorithmic randomness. We present one possibility and argue that it cannot occupy the same position in the study of algorithmic randomness that the Church-Turing thesis does in computability theory. We begin by observing that some evidence comparable to that for the Church-Turing thesis does exist for this statement: in particular, there are other reasonable formalizations of the intuitive concept of randomness that lead to the same class of random sequences (the Martin-Löf random sequences). However, we consider three properties that we would like a random sequence to satisfy and find that the Martin-Löf random sequences do not necessarily possess these properties to a greater degree than other types of random sequences, and we further argue that there is no more appropriate version of the Church-Turing thesis for algorithmic randomness. This suggests that consensus around a version of the Church-Turing thesis in this context is unlikely.

Supported in part by Simons Foundation Collaboration Grant #420806.

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Notes

  1. 1.

    We note that there are other randomness notions that also exhibit computational weakness, e.g., weak 2-randomness. However, we discuss difference randomness here because the difference random sequences can be identified as the Martin-Löf random sequences that are computationally weak in these two standard senses.

  2. 2.

    The reader may have noted that we are working in a general computable probability space rather than the Cantor space. This is possible because any computable probability space is isomorphic to the Cantor space in every relevant way and our notions of randomness transfer naturally [13].

  3. 3.

    There is a subtlety in this result in that an incomputable function may be computable as a vector, hence the “essentially."

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Authors and Affiliations

  1. Hofstra University, Hempstead, NY, 11590, USA

    Johanna N. Y. Franklin

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  1. Johanna N. Y. Franklin
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Correspondence to Johanna N. Y. Franklin .

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Editors and Affiliations

  1. University of Lille, Lille, France

    Liesbeth De Mol

  2. Vakgroep Wiskunde, Ghent University, Ghent, Belgium

    Andreas Weiermann

  3. University of Göttingen, Göttingen, Germany

    Florin Manea

  4. Ghent University, Ghent, Belgium

    David Fernández-Duque

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Franklin, J.N.Y. (2021). A Church-Turing Thesis for Randomness?. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_20

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  • DOI: https://doi.org/10.1007/978-3-030-80049-9_20

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