A generalization of Chaitin's halting probability $\Omega$ and halting self-similar sets

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 Ω is generalized to Ω 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 selfsimilar 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 Sierpiński gasket. Knowledge of the property of Ω 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.

Kolmogorov Complexity and Algorithmic Randomness

“…163 Пусть | эффективно открытое подмножество отрезка, имеющее меру меньше 1 и содержащее все неслучайные последовательности (скажем, одно из множеств универсального теста Мартин-Лёфа). Докажите, что мера является перечислимым снизу случайным числом.…”

“…[163,130]). Будем говорить, что множество ⊂˙является эф-фективно -нулевым (для данного > 0), если существует алгоритм, который по рациональному > 0 указывает покрытие множества интервалами, у которых сумма -степеней мер не превосходит .…”

A statistical mechanical interpretation of algorithmic information theory III: composite systems and fixed points

“…In the works [11], [12], we generalized the notion of the randomness of a real so that the degree of the randomness, which is often referred to as the partial randomness recently [2], [9], [3], can be characterized by a real T with 0 ≤ T ≤ 1 as follows.…”

“…Theorem 5 (properties of Z(T ) and F (T ), [11], [12], [14]). Let V be an optimal computer, and let T ∈ R.…”

On universal computably enumerable prefix codes

“…The more general fact that w∈W r −α·H U (w) diverges for α < 1 and arbitrary infinite c.e. W ⊆ X * was derived in Equation (49) of Tadaki (2002). For the sake of completeness, we prove it as Lemma 15 below.…”