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Randomness and halting probabilities

Published online by Cambridge University Press:  12 March 2014

VeróNica Becher
Affiliation:
Departamento De Computación, Facultad De Ciencias Exactas Y Naturales, Universidad De Buenos Aires, Argentina and Conicet, Argentina, E-mail: vbecher@dc.uba.ar
Santiago Figueira
Affiliation:
Departamento De Computación, Facultad De Ciencias Exactas Y Naturales, Universidad De Buenos Aires, Argentina, E-mail: sflgueir@dc.uba.ar
Serge Grigorieff
Affiliation:
Liafa, Université Paris 7 & CNRS, France, E-mail: seg@liafa.jussieu.fr
Joseph S. Miller
Affiliation:
Department of Mathematics, University of Connecticut, USA, E-mail: joseph.miller@math.uconn.edu

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 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2006

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References

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