Opacity complexity of automatic sequences. The general case

TL;DR

This paper introduces the concept of opacity complexity to quantify the complexity of automatic sequences, providing algorithms and calculations for various well-known sequences.

Contribution

It proposes a new measure called opacity complexity for automatic sequences and develops an algorithm to compute it, with applications to several classical sequences.

Findings

Opacity complexity is computable for various automatic sequences.

The paper provides explicit opacity complexity values for sequences like Thue-Morse and Golay-Shapiro.

The notion helps understand the structural complexity of automatic sequences.

Abstract

In this work we introduce a new notion called opacity complexity to measure the complexity of automatic sequences. We study basic properties of this notion, and exhibit an algorithm to compute it. As applications, we compute the opacity complexity of some well-known automatic sequences, including in particular 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, the Tower of Hanoi sequence, and so on.