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From index sets to randomness in ∅n: random reals and possibly infinite computations part II

  • Verónica Becher (a1) and Serge Grigorieff (a2)

We obtain a large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle ∅(n−1)) à la Chaitin. Any such real is defined as the probability that a universal monotone Turing machine performing possibly infinite computations on infinite (resp. finite large enough, resp. finite self-delimited) inputs produces an output in a given set . In particular, we develop methods to transfer many-one completeness results of index sets to n-randomness of associated probabilities.

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[4] V. Becher and S. Grigorieff , Recursion and topology on 2≤ω for possibly infinite computations, Theoretical Computer Science, vol. 322 (2004), pp. 85136.

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[8] C.S. Calude , P.H. Hertling , and B. Khoussainov Y. Wang , Recursively enumerable reals and Chaitin Ω numbers, Stacs 98 (Paris, 1998), Lecture Notes in Computer Science, vol. 1373, Springer-Verlag, 1998, pp. 596606.

[9] G. Chaitin , A theory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329340, Available on Chaitin's home page.

[11] G. Hjorth and A. Nies , Randomness via effective descriptive set theory, The Journal of the London Mathematical Society, vol. 75 (2007), no. 2, pp. 495508.

[12] G. Kreisel , J.R. Shoenfield , and H. Wang , Number theoretic concepts and recursive well-orderings, Archivfur math. Logik und Grundlagenforschung, vol. 5 (1960), pp. 4264.

[16] V.L. Selivanov , Hierarchies in φ-spaces and applications. Mathematical Logic Quaterly, vol. 51 (2005), no. 1, pp. 4561.

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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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