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A Generalization of Chaitin's Halting Probability Omega and Halting Self-Similar Sets | | Kohtaro Tadaki
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2 Dec 2002 | | Journal: | Hokkaido Mathematical Journal, Vol. 31, No. 1, February 2002, 219-253 | | Subject: | Chaotic Dynamics; Computational Complexity | nlin.CD cs.CC | | Abstract: | We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D>0. Chaitin’s halting probability Omega is generalized to Omega^D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree of randomness of each point in Euclidean space through its base-two expansion. It is then shown that the maximum value of such a degree of randomness provides the Hausdorff dimension of a self-similar set that is computable in a certain sense. The class of such self-similar sets includes familiar fractal sets such as the Cantor set, von Koch curve, and Sierpinski gasket. Knowledge of the property of Omega^D allows us to show that the self-similar subset of [0,1] defined by the halting set of a universal algorithm has a Hausdorff dimension of one. | | Source: | arXiv, nlin.CD/0212001 | | Services: | Forum | Review | PDF | Favorites | |
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