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).
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