Automaticity of spacetime diagrams generated by cellular automata on commutative monoids
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915798606413824 |
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| author | Nesme, Vincent |
| author_facet | Nesme, Vincent |
| contents | It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. It has been shown that such patterns can occur when the alphabet is endowed with the structure of an Abelian group, provided the cellular automaton is a morphism with respect to this structure and the initial configuration has finite support. The spacetime diagram then has a property related to k-automaticity. We show that these conditions can be relaxed: the Abelian group can be a commutative monoid, the initial configuration can be k-automatic, and the spacetime diagrams still exhibit the same regularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13062 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Automaticity of spacetime diagrams generated by cellular automata on commutative monoids Nesme, Vincent Discrete Mathematics Formal Languages and Automata Theory Dynamical Systems 11B85, 37B15 F.1.1 It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. It has been shown that such patterns can occur when the alphabet is endowed with the structure of an Abelian group, provided the cellular automaton is a morphism with respect to this structure and the initial configuration has finite support. The spacetime diagram then has a property related to k-automaticity. We show that these conditions can be relaxed: the Abelian group can be a commutative monoid, the initial configuration can be k-automatic, and the spacetime diagrams still exhibit the same regularity. |
| title | Automaticity of spacetime diagrams generated by cellular automata on commutative monoids |
| topic | Discrete Mathematics Formal Languages and Automata Theory Dynamical Systems 11B85, 37B15 F.1.1 |
| url | https://arxiv.org/abs/2207.13062 |