Exact expressions for some randomness tests

  • Péter Gács
Vorträge (In Alphabetischer Reihenfolge)
Part of the Lecture Notes in Computer Science book series (LNCS, volume 67)


For a computable probability distribution P over the set of infinite binary sequences Martin-Löf defined a test d(x|P) measuring the degree of nonrandomness of the sequence x with respect of P. We give some expressions in terms of Kolmogorov's and other complexities of the initial segments of x whose difference from d(x|P) is bounded by a constant.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Martin-Löf, Per: The definition of random sequences. Information and Control 6 (1966) 602–619Google Scholar
  2. [2]
    Zvonkin, A.K., Levin, L.A.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russ. Math. Surv. 25/6 (1970) 83Google Scholar
  3. [3]
    Levin, L.A.: On the notion of a random sequence. Soviet Math. Dokl. 14 (1973) 1413Google Scholar
  4. [4]
    Schnorr, C.P.: Process complexity and effective random tests. J. Comput. Syst. Sci. 7 (1973) 376Google Scholar
  5. [5]
    Kolmogorov, A.N.: Three approaches to the quantitative definition of information, Prob. Info. Transmission 1/1 (1965) 1Google Scholar
  6. [6]
    Levin, L.A.: Some theorems on the algorithmic approach to Probability Theory and Information Theory (Ph.D.thesis, 1971)Google Scholar
  7. [7]
    Chaitin, G.J.: A theory of program-size formally identical to Information Theory, J.A.C.M. 22 (1975) 329Google Scholar
  8. [8]
    Chaitin, G.J.: Algorithmic Information Theory, IBM J. REs. Dev. (July 1977) 350–359Google Scholar
  9. [9]
    Schnorr, C.P.: Unpublished manuscriptGoogle Scholar
  10. [10]
    Martin-Löf, P.: Algorithmen und zufällige Folgen. Lecture Notes, University of Erlangen, 1966Google Scholar
  11. [11]
    Levin, L.A.: Uniform tests of randomness, Soviet Math. Doklady 17 (1976) 337Google Scholar
  12. [12]
    Gács, P.: On the symmetry of algorithmic information. Soviet Math. Dokl. 15 (1974) 1477–1480, Corrections ibid. No. 6, V.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • Péter Gács

There are no affiliations available

Personalised recommendations