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Journal Article

# Randomness and Halting Probabilities

Verónica Becher, Santiago Figueira, Serge Grigorieff and Joseph S. Miller
The Journal of Symbolic Logic
Vol. 71, No. 4 (Dec., 2006), pp. 1411-1430
Stable URL: http://www.jstor.org/stable/27588521
Page Count: 20

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## Abstract

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 \$\Sigma _{n}^{0}\$-complete or \$\Pi _{n}^{0}\$-complete for some n ≥ 2. However, for n ≥ 2, ΩU[X] is not n-random when X is \$\Sigma _{n}^{0}\$ or \$\Pi _{n}^{0}\$ Nevertheless, there exists \$\Delta _{n+1}^{0}\$ sets such that ΩU[X] is n-random. There are \$\Delta _{2}^{0}\$ sets X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is \$\Delta _{n+1}^{0}\$ and \$\Sigma _{n}^{0}\$-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 ⊆ 2<ω} 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 ⊆ 2<ω 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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