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Computational randomness and lowness*

  • Sebastiaan A. Terwijn (a1) and Domenico Zambella (a2)

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0′. This contrasts with a result of Kučera and Terwijn [5] on sets that are low for the class of Martin-Löf random reals.

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Research supported by the Netherlands Foundation for Scientific Research (NWO) Project PGS 22-262. Most of this research was done while the authors were working at the University of Amsterdam.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] K. Ambos-Spies and A. Kučera , Randomness in computability theory, Computability theory and its applications: Current trends and open problems ( P. Cholak , S. Lempp , M. Lerman , and R. A. Shore , editors), Contemporary Mathematics, vol. 257, AMS, Providence RI, 2000, pp. 114.

[4] A. Kučera , Measure, Π10-classes, and complete extensions of PA, Recursion Theory Week 1984 ( H.-D. Ebbinghaus , G. H. Müller , and G. E. Sacks , editors), Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, 1985, pp. 245259.

[7] P. Martin-Löf , The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602619.

[8] W. Miller and D. A. Martin , The degrees of hyperimmune sets, Z. Math. Logik Grundlagen Math, vol. 14 (1968), pp. 159166.

[10] J. Raisonnier , A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel Journal of Mathematics, vol. 48 (1984), pp. 4856.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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