Systems of Strings with High Mutual Complexity

  • Authors:

  • M. V. Vyugin

  • Affiliations:

  • Royal Holloway University of London. M.V. Lomonosov Moscow State University misha@cs.rhul.ac.uk

  • Venue:
  • Problems of Information Transmission
  • Year:
  • 2003

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Abstract

Consider a binary string x0 of Kolmogorov complexity K(x0) ≥ n. The question is whether there exist two strings x1 and x2 such that the approximate equalities K(xi ∣ xj) ≈ n and K(xi ∣ xj, xk) ≈ n hold for all 0 ≤ i, j, k ≤ 2, i ≠ j ≠ k, i ≠ k. We prove that the answer is positive if we require the equalities to hold up to an additive term O(log K(x0)). It becomes negative in the case of better accuracy, namely, O(log n).