The authors introduce a new notion called opacity complexity to measure the complexity of automatic sequences. They study the basic properties of this notion and provide an algorithm to compute it.
The key highlights and insights are:
The opacity of a finite k-automaton A measures the intrinsic noise produced by the default of A. It is defined as the supremum over all infinite words σ of the infimum over all output functions o of the distance between o(A σ) and σ.
The opacity complexity of a k-automatic sequence u is defined as the supremum over all finite k-automata in AUTk(u) of the opacity of the automaton, normalized by the maximum opacity Mk.
The authors show that the supremum in the definition of opacity complexity can be achieved by a special finite k-automaton Au, called the intrinsic finite k-automaton of u.
The authors provide a detailed characterization of transparent and opaque finite k-automata, and use this to compute the opacity complexity of several classical automatic sequences, including constant sequences, purely periodic sequences, the Thue-Morse sequence, the period-doubling sequence, the Golay-Shapiro(-Rudin) sequence, the paperfolding sequence, the Baum-Sweet sequence, and the Tower of Hanoi sequence.
The authors indicate that in a subsequent work, they will investigate the opacity complexity attached to a different comparison method, the quadratic semi-norm, which may lead to different results for certain special automatic sequences.
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by J.-P. Allouc... at arxiv.org 04-23-2024
https://arxiv.org/pdf/2404.13601.pdfDeeper Inquiries