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Randomness and halting probabilities

Published online by Cambridge University Press:  12 March 2014

VeróNica Becher ,
Santiago Figueira ,
Serge Grigorieff  and
Joseph S. Miller
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
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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 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 ⊆ 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1411 - 1430
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Becher, V. and Grigorieff, S., Random reals and possibly infinite computations. Part I: randomness in ∅′, this Journal, vol. 70 (2005), no. 3, pp. 891–913.Google Scholar
[2]Becher, V. and Grigorieff, S., Random reals and possibly infinite computations. Part II: From index sets to higher order randomness, In preparation.Google Scholar
[3]Calude, C., Hertling, P., Khoussainov, B., and Wang, Y., Recursively enumerable reals and Chaitin Omega numbers, Theoretical Computer Science, vol. 255 (2001), no. 1–2, pp. 125–149.CrossRefGoogle Scholar
[4]Chaitin, G. J., A theory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329–340.CrossRefGoogle Scholar
[5]Chaitin, G. J., Algorithmic information theory, Cambridge University Press, 1988.Google Scholar
[6]Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity, to appear.Google Scholar
[7]Downey, R., Hirschfeldt, D., and Nies, A., Randomness, computability and density, SIAM Journal on Computing, vol. 31 (2002), pp. 1169–1183.CrossRefGoogle Scholar
[8]Figueira, S., Stephan, F., and Wu, G., Randomness and universal machines, CCA 2005, Second International Conference on Computability and Complexity in Analysis, Kyoto, Fernuniversität Hagen, Informatik Berlchte, vol. 326, 2005, pp. 103–116.Google Scholar
[9]Kučera, A. and Slaman, T. A., Randomness and recursive enumerability, SIAM Journal on Computing, vol. 31 (2001), pp. 199–211.CrossRefGoogle Scholar
[10]Odifreddi, P., Classical recursion theory, North-Holland, 1990.Google Scholar
[11]Junior, H. Rogers, Theory of recursive functions and effective computability, McGraw Hill, New York, 1968.Google Scholar
[12]Solovay, R. M., Draft of paper (or series of papers) on Chaitin's work, done for the most part during the period of Sept.–Dec. 1974, unpublished manuscript, IBM Thomas Watson Research Center, Yorktown Heights, NY, 215 pages.Google Scholar