A Generalization of Chaitin's Halting Probability \Omega and Halting Self-Similar Sets
<|reference_start|>A Generalization of Chaitin's Halting Probability \Omega and Halting Self-Similar Sets: 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.<|reference_end|>
@article{tadaki2002a,
    title={A Generalization of Chaitin's Halting Probability \Omega and Halting
  Self-Similar Sets},
    author={Kohtaro Tadaki},
    journal={Hokkaido Mathematical Journal, Vol. 31, No. 1, February 2002,
  219-253},
    year={2002},
    doi={10.14492/hokmj/1350911778},
    archivePrefix={arXiv},
    eprint={nlin/0212001},
    primaryClass={nlin.CD cs.CC}
}