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Highness properties close to PA completeness

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Abstract

Suppose we are given a computably enumerable object. We are interested in the strength of oracles that can compute an object that approximates this c.e. object. It turns out that in many cases arising from algorithmic randomness or computable analysis, the resulting highness property is either close to, or equivalent to being PA complete. We examine, for example, majorizing a c.e. martingale by an oracle-computable martingale, computing lower bounds for two variants of Kolmogorov complexity, and computing a subtree of positive measure with no dead-ends of a given \(\prod _1^0\) tree of positive measure. We separate PA completeness from the latter property, called the continuous covering property. We also separate the corresponding principles in reverse mathematics.

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Victoria University of Wellington, Po Box 600, Wellington, 6140, New Zealand

    Noam Greenberg

  2. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI, 53706, USA

    Joseph S. Miller

  3. Department of Computer Science, University of Auckland, Private bag 92019, Auckland, 1142, New Zealand

    André Nies

Authors
  1. Noam Greenberg
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  2. Joseph S. Miller
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  3. André Nies
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Corresponding author

Correspondence to Noam Greenberg.

Additional information

The research in this paper was commenced when the authors participated in the Buenos Aires Semester in Computability, Complexity and Randomness, 2013. Greenberg and Nies were partially supported by a Marsden grant of the Royal Society of New Zealand. Miller was initially supported by the National Science Foundation under grant DMS-1001847; he is currently supported by grant #358043 from the Simons Foundation.

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Greenberg, N., Miller, J.S. & Nies, A. Highness properties close to PA completeness. Isr. J. Math. 244, 419–465 (2021). https://doi.org/10.1007/s11856-021-2200-7

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  • DOI: https://doi.org/10.1007/s11856-021-2200-7

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