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. In our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425-434, 2008, we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T , such as free energy F (T ), energy E(T ), and statistical mechanical entropy S(T ), into the theory. These quantities are real functions of real argument T > 0. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity. Namely, the computability of the value of partition function Z(T ) gives a sufficient condition for T ∈ (0, 1) to be a fixed point on partial randomness. In this paper, we show that the computability of each of all the thermodynamic quantities above gives the sufficient condition also. Moreover, we show that the computability of F (T ) gives completely different fixed points from the computability of Z(T ).

We develop a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities, such as free energy, energy, statistical mechanical entropy, and specific heat, into algorithmic information theory. We investigate the properties of these quantities by means of program-size complexity from the point of view of algorithmic randomness. It is then discovered that, in the interpretation, the temperature plays a role as the compression rate of the values of all these thermodynamic quantities, which include the temperature itself. Reflecting this self-referential nature of the compression rate of the temperature, we obtain fixed point theorems on compression rate.

We proposed the concept, piece in hand (soldiers in hand) matrix and have developed the framework based on the concept so far. The piece in hand matrix is a general concept which can be applicable to any type of multivariate public key cryptosystems to enhance their security. In this paper, we make improvements in the PH matrix method as follows. (i) In the PH matrix method, an arbitrary number of additional variables can be introduced to the random polynomial term in the public key, which is eliminated by the multiplication of the PH matrix to the public key in the decryption. Thus these additional variables enables the public key to have more than one solution, and therefore increases the difficulty to solve the public key. We show, in an experimental manner, that the PH matrix method improved in this way is secure even against the Gröbner basis attack. (ii) In the nonlinear PH matrix method proposed previously, the degree of polynomials in the public key is more than two, and this may cause an undesirable increase in the length of the public key. In this paper, we propose a nonlinear PH matrix method, where the degree of the public key is kept the same as the degree of the public key of the original cryptosystem, which is normally two.

In this paper we develop a statistical mechanical interpretation of the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the context of noiseless source coding. Especially, it is discovered that the temperature 1 corresponds to the average codeword length of an instantaneous code in this statistical mechanical interpretation of noiseless source coding scheme. This correspondence is also verified by the investigation using box-counting dimension. Using the notion of temperature and statistical mechanical arguments, some information-theoretic relations can be derived in the manner which appeals to intuition.