A one-dimensional cellular automaton is a discrete dynamical system where a sequence of symbols evolves synchronously according to a local update rule. We discuss simple update rules that make the automaton perform multiplications of numbers by a constant. If the constant and the number base are selected suitably the automaton becomes a universal pattern generator: all finite strings over its state alphabet appear from a finite seed. In particular we consider the automata that multiply by constants 3 and 3/2 in base 6. We discuss the connections of these automata to some difficult open questions in number theory, and we pose several further questions concerning pattern generation in cellular automata.
@InProceedings{kari:LIPIcs.RTA.2013.1, author = {Kari, Jarkko}, title = {{Pattern Generation by Cellular Automata}}, booktitle = {24th International Conference on Rewriting Techniques and Applications (RTA 2013)}, pages = {1--3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-53-8}, ISSN = {1868-8969}, year = {2013}, volume = {21}, editor = {van Raamsdonk, Femke}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.1}, URN = {urn:nbn:de:0030-drops-40490}, doi = {10.4230/LIPIcs.RTA.2013.1}, annote = {Keywords: cellular automata, pattern generation, Z-numbers} }
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