For a string (xn), generated by sampling a probability distribution P(xn), we have already suggested the ideal code length — logP(xn) to serve as its complexity, the Shannon complexity, with the justification that its mean is for large alphabets a tight lower bound for the mean prefix code length. The problem, of course, arises that this measure of complexity depends very strongly on the distribution P, which in the cases of interest to us is not given. Nevertheless, we feel intuitively that a measure of complexity ought to be linked with the ease of its description. For instance, consider the following three types of data strings of length n = 20, where the length actually ought to be taken large to make our point:
  1. 1.


  2. 2.


  3. 3.

    generate a string by flipping a coin 20 times



Binary String Recursive Function Code Length Kolmogorov Complexity Regular Feature 
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© Springer Science+Business Media, LLC 2007

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