### Randomness and halting probabilities

Verónica Becher, Santiago Figueira, Serge Grigorieff, and Joseph S. Miller
Source: J. Symbolic Logic Volume 71, Issue 4 (2006), 1411-1430.

#### 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 Σ⁰n-complete or Π⁰n-complete for some n≥2.
• However, for n≥2, ΩU[X] is not n-random when X is Σ⁰n or Π⁰n. Nevertheless, there exists Δ⁰n+1 sets such that ΩU[X] is n-random.
• There are Δ⁰₂ sets X such that ΩU[X] is rational. Also, for every n≥1, there exists a set X which is Δ⁰n+1 and Σ⁰n-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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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1164060463
Digital Object Identifier: doi:10.2178/jsl/1164060463
Zentralblatt MATH identifier: 1152.03038
Mathematical Reviews number (MathSciNet): MR2275867

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