Every Sequence Is Decompressible from a Random One

  • David Doty
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)


Kučera and Gács independently showed that every infinite sequence is Turing reducible to a Martin-Löf random sequence. We extend this result to show that every infinite sequence S is Turing reducible to a Martin-Löf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. We show that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, we give a new characterization of constructive dimension in terms of Turing reduction compression ratios.


Constructive dimension Kolmogorov complexity Turing reduction compression martingale random sequence 


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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David Doty
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

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