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A construction of simple and smaller-state real-time generator for exponential sequences

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Abstract

A model of cellular automata (\({\mathrm {CA}}\)) is considered to be a well-studied non-linear model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in a synchronous manner according to a uniform local rule. A sequence generation problem on the \({\mathrm {CA}}\)s was studied and many scholars proposed real-time sequence generation algorithms for a variety of non-regular sequences such as \(\{2^n \,|\,n = 1, 2, 3,\ldots \}\), prime, and Fibonacci sequences. In this paper, we show that sequence \(\{ k^n | n=1, 2, 3, \ldots \}\) can be generated in real-time by a k-state \({\mathrm {CA}}\), when \(k \ge 3\), and give a mathematical proof of the correctness of the implementation.

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Authors and Affiliations

  1. Osaka Electro-Communication University, 18-8 Hatsu-cho, Neyagawa-shi, Osaka, 572-8530, Japan

    Naoki Kamikawa & Hiroshi Umeo

Authors
  1. Naoki Kamikawa
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  2. Hiroshi Umeo
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Correspondence to Naoki Kamikawa.

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This work was presented in part at the 24th International Symposium on Artificial Life and Robotics, Beppu, Oita, January 23–25, 2019.

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Kamikawa, N., Umeo, H. A construction of simple and smaller-state real-time generator for exponential sequences. Artif Life Robotics 25, 64–72 (2020). https://doi.org/10.1007/s10015-019-00569-3

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