GENERATIVE COMPLEXITY AND RANDOMNESS OF FINITE SEQUENCE OF SYMBOLS

Abstract

Here proposed are complexity measures of finite sequences of symbols, based on finite automata. Basic properties of these measures are demonstrated. The relation between the complexity for generating a sequence and the randomness of the generated sequence is also discussed. First, the notion of A-complexity is defined and characterized using ultimately periodic sequences (Theorem 1). A refined measure, F-complexity, is then introduced. It is shown that highly random sequences have large F-complexities (Theorem 2), but the converse is not always true (Theorem 3). Finally, the c-complexity is proposed to remedy this shortcoming of F-complexity. It includes as special cases both A-complexity and F-complexity. It is shown that certain sequences with high c-complexities, complete periodic sequences, are equidistributed (Theorem 4).