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.

The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion of probability is not established yet. In this paper, based on the toolkit of algorithmic randomness we present an operational characterization of the notion of probability, called an ensemble. Algorithmic randomness, also known as algorithmic information theory, is a field of mathematics which enables us to consider the randomness of an individual infinite sequence. We use the notion of Martin-Löf randomness with respect to Bernoulli measure to present the operational characterization. As the first step of the research of this line, in this paper we consider the case of finite probability space, i.e., the case where the sample space of the underlying probability space is finite, for simplicity. We give a natural operational characterization of the notion of conditional probability in terms of ensemble, and give equivalent characterizations of the notion of independence between two events based on it. Furthermore, we give equivalent characterizations of the notion of independence of an arbitrary number of events/random variables in terms of ensembles. In particular, we show that the independence between events/random variables is equivalent to the independence in the sense of van Lambalgen's Theorem, in the case where the underlying finite probability space is computable. In the paper we make applications of our framework to information theory and cryptography in order to demonstrate the wide applicability of our framework to the general areas of science and technology.

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H(s) of a given finite binary string s. In the standard way, H(s) is defined as the length of the shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H(s) is defined as − log 2 m(s) without reference to the concept of program-size.Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed.In what follows, we introduce an operator versionĤ(s) of H(s) in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties.

Abstract. Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach., vol. 22, pp. 329-340, 1975] introduced Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the base-two expansion of Ω solve the halting problem of the optimal computer for all binary inputs of length at most n. In the present paper we investigate this property from various aspects. We consider the relative computational power between the base-two expansion of Ω and the halting problem by imposing the restriction to finite size on both the problems. It is known that the base-two expansion of Ω and the halting problem are Turing equivalent. We thus consider an elaboration of the Turing equivalence in a certain manner.

Abstract. A real α is called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real α can be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin Ω number, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rationals which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real T ∈ (0, 1], where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. As a result, we present ten equivalent characterizations of the partial randomness of an r.e. real. The resultant characterizations of partial randomness are powerful and have many important applications. One of them is to present equivalent characterizations of the dimension of an individual r.e. real. The equivalence between the notion of Hausdorff dimension and compression rate by program-size complexity (or partial randomness) has been established at present by a series of works of many researchers over the last two decades. We present ten equivalent characterizations of the dimension of an individual r.e. real.