Mutual Dimension

Authors Adam Case, Jack H. Lutz

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Adam Case
Jack H. Lutz

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Adam Case and Jack H. Lutz. Mutual Dimension. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 116-126, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


We define the lower and upper mutual dimensions mdim(x:y) and Mdim(x:y) between any two points x and y in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by x and y. We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that, if f : R^m -> R^n is computable and Lipschitz, then the inequalities mdim(f(x):y) <= mdim(x:y) and Mdim(f(x):y) <= Mdim(x:y) hold for all x \in R^m and y \in R^t. We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.
  • computable analysis
  • data processing inequality
  • effective fractal dimensions
  • Kolmogorov complexity
  • mutual information


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