Divisibility and State−Complexity

A sequence a = Han L is B−automatic if it can be recognized by a finite state machine in the sense that, on input n, the machine outputs an . Here n is usually written in standard base B notation, though other numeration systems are also considered. A typical example is the Morse−Thue sequence Htn L which is usually defined in terms of iterated morphisms. However, there is an alternative characterization that shows that tn is the digit−sum of the binary expansion of n modulo 2. Thus, the Morse−Thue sequence is 2−automatic. The study of automaticity has lately attracted a lot of attention, see the excellent book by Allouche and Shallit [AS].