Based on program-size complexity, a logical basis for information theory and probability theory has been proposed by A. N. Kolmogorov[1]. The aim of this paper is to further strengthen this logical basis and make it more perfect. First, for the general case of computable probability distributions, sufficient and necessary conditions are given for an infinite sequence x is-an-element-of A-infinity to be a Martin-lof (M. L.) infinite random sequence of a computable probability distribution. These sufficient and necessary conditions give a complexity-based definition of an infinite random sequence which is equivalent to P. Martin-lof's statistical definition of the concept of randomness. Consequently, a common complexity-based theory of finite and infinite random sequences is established. Finally, inequalities between Chaitin complexity and Shannon information content of a single event are given, and asymptotically equivalent relationships between them are also presented.
