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Opacity Complexity of Automatic Sequences: A General Approach

Core Concepts
The opacity complexity of an automatic sequence measures the intrinsic noise or distortion between the sequence and the words defined by the k-ary expansion of integers.
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.

Key Insights Distilled From

by J.-P. Allouc... at 04-23-2024
Opacity complexity of automatic sequences. The general case

Deeper Inquiries

How does the opacity complexity of an automatic sequence relate to other complexity measures, such as subword complexity or transducer degrees

The opacity complexity of an automatic sequence provides a unique perspective on the complexity of the sequence by measuring the intrinsic noise produced by the default of the finite automaton generating the sequence. In contrast, subword complexity focuses on counting the number of different subwords of a certain length in the sequence, providing insight into the structural complexity of the sequence. On the other hand, transducer degrees compare two infinite words based on their transformation under a sequential finite-state transducer, offering a way to compare the complexity of different sequences in terms of their transformations. While subword complexity and transducer degrees focus on structural and transformational aspects, respectively, the opacity complexity delves into the inherent noise and distortion between the sequence and the words defined by the k-ary expansion of n's, providing a distinct measure of complexity.

Can the notion of opacity complexity be extended to other types of sequences beyond automatic sequences, such as morphic sequences or Sturmian sequences

The notion of opacity complexity can potentially be extended to other types of sequences beyond automatic sequences, such as morphic sequences or Sturmian sequences. By adapting the concept of opacity complexity to these sequences, it could offer insights into their inherent noise and distortion, providing a new measure of complexity specific to these sequence types. For morphic sequences, which exhibit self-similarity under iteration, the opacity complexity could shed light on the level of noise introduced during the generation process. Similarly, for Sturmian sequences, known for their rich combinatorial properties, opacity complexity could reveal unique characteristics related to their generation and structure.

What are the potential applications of the opacity complexity measure in areas like cryptography, information theory, or dynamical systems

The opacity complexity measure holds significant potential for various applications in cryptography, information theory, and dynamical systems. In cryptography, understanding the inherent noise and distortion in sequences can aid in developing secure encryption algorithms that are resilient to potential attacks exploiting these characteristics. By incorporating opacity complexity into cryptographic protocols, it may enhance the security and robustness of encryption schemes. In information theory, opacity complexity can be utilized to quantify the complexity of data streams or communication channels, providing a novel metric for assessing the information content and processing requirements. In dynamical systems, opacity complexity could offer insights into the complexity of dynamic processes, helping to analyze and predict system behavior based on the level of intrinsic noise and distortion present in the sequences generated by these systems.