A generalization of Chaitin's halting probability $\Omega$ and halting self-similar sets
Kohtaro Tadaki · Hokkaido Mathematical Journal · 2002
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 Sierpin'ski 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.