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<p>The main obstacle in the quest for non-uniform cellular automata that meet the often desired property of number conservation is the vast size of the search space, going far beyond the capabilities of today’s computers. In this paper, we expound the construction of a directed graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11047_2025_10015_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Π</mi> </math></EquationSource> </InlineEquation> related to the set of all number-conserving non-uniform one-dimensional binary cellular automata with radius one and half (i.e., the neighborhood of a cell consists of four cells). We show that there is a one-to-one correspondence between the set of all such cellular automata on a&#xa0;finite grid with <i>n</i> cells and the set of all length-<i>n</i> closed directed walks in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11047_2025_10015_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Π</mi> </math></EquationSource> </InlineEquation>. This provides us with a powerful tool to investigate non-uniform cellular automata of this type.</p>

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A directed graph allowing for the exploration of the set of number-conserving non-uniform one-dimensional binary cellular automata with radius one and half

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月5日

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  • Barbara Wolnik,
  • Maciej Dziemiańczuk,
  • Bartosz Makuracki,
  • Bernard De Baets

摘要

The main obstacle in the quest for non-uniform cellular automata that meet the often desired property of number conservation is the vast size of the search space, going far beyond the capabilities of today’s computers. In this paper, we expound the construction of a directed graph \(\Pi \) Π related to the set of all number-conserving non-uniform one-dimensional binary cellular automata with radius one and half (i.e., the neighborhood of a cell consists of four cells). We show that there is a one-to-one correspondence between the set of all such cellular automata on a finite grid with n cells and the set of all length-n closed directed walks in \(\Pi \) Π . This provides us with a powerful tool to investigate non-uniform cellular automata of this type.