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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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Randomness and Halting Probabilities
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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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