Automatic complexity of Fibonacci and Tribonacci words

10/14/2020

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by   Bjørn Kjos-Hanssen, et al.

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For a complexity function C, the lower and upper C-complexity rates of an infinite word 𝐱 are C(𝐱)=lim inf_n→∞C(𝐱↾ n)/n, C(𝐱)=lim sup_n→∞C(𝐱↾ n)/n respectively. Here 𝐱↾ n is the prefix of x of length n. We consider the case C=A_N, the nondeterministic automatic complexity. If these rates are strictly between 0 and 1/2, we call them intermediate. Our main result is that words having intermediate A_N-rates exist, viz. the infinite Fibonacci and Tribonacci words.