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Randomness and halting probabilities

  • VeróNica Becher (a1), Santiago Figueira (a2), Serge Grigorieff (a3) and Joseph S. Miller (a4)

We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows:

• ΩU[X] is random whenever X is Σn0-complete or Πn0-complete for some n ≥ 2.

• However, for n ≥ 2, ΩU[X] is not n-random when X is Σn0 or Πn0. Nevertheless, there exists Δn+10 sets such that ΩU[X] is n-random.

• There are Δ20 sets X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is Δn+10 and Σn0-hard such that ΩU[X] is not random.

We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2ω recursive in ∅′ ⊕ r, such that ΩU[X] = r.

The same questions are also considered in the context of infinite computations, and lead to similar results.

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[3] C. Calude , P. Hertling , B. Khoussainov , and Y. Wang , Recursively enumerable reals and Chaitin Omega numbers, Theoretical Computer Science, vol. 255 (2001), no. 1–2, pp. 125149.

[4] G. J. Chaitin , A theory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329340.

[7] R. Downey , D. Hirschfeldt , and A. Nies , Randomness, computability and density, SIAM Journal on Computing, vol. 31 (2002), pp. 11691183.

[9] A. Kučera and T. A. Slaman , Randomness and recursive enumerability, SIAM Journal on Computing, vol. 31 (2001), pp. 199211.

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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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