Normal Sequences with Non-Maximal Automatic Complexity

Abstract

This paper examines Automatic Complexity, a complexity notion introduced by Shallit and Wang in 2001 [28]. We demonstrate that there exists a normal sequence T such that I(T ) = 0 and S(T ) ≤ 1/2, where I(T ) and S(T ) are the lower and upper automatic complexity rates of T respectively. We furthermore show that there exists a Champernowne sequence C, i.e. a sequence formed by concatenating all strings of length 1 followed by concatenating all strings of length 2 and so on, such that S(C) ≤ 2/3.